karsten fritzsch [email protected] · 2018-10-11 · 2016–17 differential geometry...

Differential Geometry

Karsten Fritzsch

[email protected]

Department of MathematicsUniversity College London

2016–17

These are lecture notes for a one-term course on Differential Geometry at the University College London,they are based on lecture notes by Rod Halburd and Daniel Grieser. Thanks for those!

Thanks to Rod for the comments and remarks on Differential Geometry and on teaching it at UCL and toDaniel for being my teacher.

Last updated: 1st February 2017

Contents

I Background Material 1I.1 Basic Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

I.2 Differentiability of Maps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

I.3 Multivariate Integration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

I.4 Ordinary Differential Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

References and Further Reading 12

Introduction 13

II The Theory of Curves 14II.1 Parameterised Curves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

II.2 The Local Theory of Space Curves . . . . . . . . . . . . . . . . . . . . . . . . . 18

II.3 The Global Theory of Plane Curves . . . . . . . . . . . . . . . . . . . . . . . . . 24

III Submanifolds 30III.1 Submanifolds of Euclidean Space . . . . . . . . . . . . . . . . . . . . . . . . . . 30

III.2 The Tangent Space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

III.3 Functions on Submanifolds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

III.4 Surfaces in 3-Dimensional Space . . . . . . . . . . . . . . . . . . . . . . . . . . . 44

IV The Fundamental Forms and Curvature 45IV.1 The First Fundamental Form . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

IV.2 Orientability and the Gauss Map . . . . . . . . . . . . . . . . . . . . . . . . . . 50

IV.3 The Second Fundamental Form . . . . . . . . . . . . . . . . . . . . . . . . . . . 51

IV.4 Principal, Gauss and Mean Curvatures . . . . . . . . . . . . . . . . . . . . . . . 53

IV.5 Calculations in Local Coordinates . . . . . . . . . . . . . . . . . . . . . . . . . . 55

V The Theorema Egregium and the Theorem of Bonnet 57V.1 Isometries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57

V.2 Vector Fields and the Covariant Derivative . . . . . . . . . . . . . . . . . . . . . 60

V.3 The Theorems of Gauss and of Bonnet . . . . . . . . . . . . . . . . . . . . . . . 66

VI Curves on Surfaces 70VI.1 Parallel Vector Fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70

VI.2 Geodesics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72

VI.3 Normal and Geodesic Curvature . . . . . . . . . . . . . . . . . . . . . . . . . . 76

VII The Theorem of Gauss-Bonnet 79VII.1 Chains in Surfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79

VII.2 The Local Theorem of Gauss-Bonnet . . . . . . . . . . . . . . . . . . . . . . . . 80

VII.3 The Global Theorem of Gauss-Bonnet . . . . . . . . . . . . . . . . . . . . . . . 84

VII.4 Applications of Gauss-Bonnet . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88

Brief Summary 91

Chapter I

Background Material

I.1 Basic Notation

Let us collect some elementary notation. First of all, there are some basic sets which we willrefer to:

N :=

positive integers, i.e., excluding 0

N0 :=

nonnegative integers, i.e., including 0

Z :=

the integers

R :=

the real numbers

C :=

the complex numbers

The symbols “:=” and “=:” always denote equalities which define the object on the side withthe colon.

Given a mapping f : A −→ B between any two sets A and B, we call A the domain and B therange of f . The set f (A) is called the image. If B′ ⊂ B, we write

f−1(B′) :=

a ∈ A∣∣ f (a) ∈ B′

and call this the preimage of B′ under f . If B′ = b contains a single element, we will usuallywrite f−1(b) := f−1(b). If A ⊂ Rn, B ⊂ Rm for some n, m ∈ N, the preimage under f of asingle point is also called a level set of f .

I.2 Differentiability of Maps

We briefly give the basic definitions for differentiability of maps f : Rn −→ Rm and state(without proof) central results which will be of importance throughout the course.

Differentiability

Definition I.1 (Differentiability). Let U ∈ Rn be open, x ∈ U and f : U −→ Rm. f is said to bedifferentiable at x if there is a linear map L : Rn −→ Rm such that

limh→o

f (x + h)− f (x)− L(h)|h| = 0 .

The linear map L is called the differential of f at x and denoted by D f∣∣x.

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2016–17 · Differential Geometry Differentiability of Maps · I.2

Letting n = m = 1, this gives the usual definition of the derivative f ′ of a function R −→ R.Please note that in this case we have D f

∣∣x(h) = f ′(x) · h, where h ∈ R.

Remark I.2. A vertical bar with a subscript x will always mean “(evaluated) at the point x”, for instanceD f∣∣x is the differential of f at x. The reason for this is that we need to distinguish between the point x

where the differential is taken and any vector h to which the differential is applied. Whenever we dealwith objects where this distinction is important, we will use this notation.

For the moment, consider a real-valued function φ : U −→ R. A concept closely related tothe differential, yet slightly simpler, is that of directional derivatives:

Definition I.3 (Directional Derivative). Let U ⊂ Rn be open, x ∈ U and φ : U −→ R. If, forh ∈ Rn, the limit

limt→0

φ(x + th)− φ(x)t

exists, we call it the directional derivative of φ at x in direction h and denote it by ∂hφ(x).

The important special case of partial derivatives is obtained by choosing as direction one of theelements of the standard basis e1, . . . , en of Rn. These are denoted by

∂φ

∂xj(x) := ∂ej φ(x) . (I.1)

(Here, and in the following, we write x = (x1, . . . , xn) with respect to this standard basis.)Collecting the partial derivatives in a vector, we define the gradient of φ at x by

∇φ(x) :=( ∂φ

∂x1(x), . . . ,

∂φ

∂xn(x))

. (I.2)

Now let us return to maps Rn −→ Rm again. Since any such map f : U −→ Rm can bewritten as a vector of component functions

f =(

f1, . . . , fm)

, (I.3)

where f j : Rn −→ R gives the j-th component of f (x) (with respect to the standard basis of Rm),we may collect all partial derivatives of all component functions in an m× n–matrix:

J f∣∣x :=

∇ f1(x)

...∇ fm(x)

=

∂ f1∂x1

(x) · · · ∂ f1∂xn

(x)...

...∂ fm∂x1

(x) · · · ∂ fm∂xn

(x)

(I.4)

This is the Jacobian1 of f at x. More generally, we denote the matrix consisting of the partialderivatives of the component functions f j1 , . . . , f jl with respect to directions ei1 , . . . , eik by

∂( f j1 , . . . , f jl )

∂(xi1 , . . . , xil )

∣∣x . (I.5)

1 Carl Gustav Jacob Jacobi, ∗ 1804 in Potsdam, † 1851 in Berlin

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(I.5) is the submatrix of J f∣∣x obtained by crossing out the rows and columns corresponding to

component functions and coordinates we did not use.The partial derivatives (and their collection in terms of the Jacobian) are an important tool to

check whether a given map is differentiable and, if so, to compute the differential.

Theorem I.4. Let U ⊂ Rn be open, x ∈ U and f : U −→ Rm.

i) If, for all 1 ≤ i ≤ n and 1 ≤ j ≤ m, the partial derivatives∂ f j∂xi

exist and are continuous onan open neighbourhood of x, then f is differentiable at x.

ii) If f is differentiable at x, then

D f∣∣x(h) =

m

∑j=1

(n

∑i=1

∂ f j

∂xi(x0) · hi

)ej =

m

∑j=1

⟨∇ f j , h

⟩ej ,

holds for all h = (h1, . . . , hn) ∈ Rn, where e1, . . . , em denotes the standard basis of Rm.

The sums in the second item of Theorem I.4 can of course be interpreted as a matrix multi-plication and the corresponding matrix is exactly the Jacobian,

D f∣∣x(h) = J f

∣∣x

h1...

hn

,

where, as before, h = (h1, . . . , hn) refers to the standard basis. Hence the Jacobian is the repres-entation of the differential with respect to the canonical bases of Rn and Rm.

Remark I.5 (The Differential vs. the Jacobian). It is important to distinguish between the linear map D f∣∣x

and the matrix J f∣∣x, the latter is just one representative for the former. For maps Rn −→ Rm,

we might have a canonical representative (the one with respect to the standard bases), but whenstudying differentials of functions between surfaces or submanifolds, there is no single distinguishedrepresentative.

Directional derivatives of a function φ : U −→ R have the same domain and range as φ,∂hφ : U −→ R. This allows us to iterate taking directional derivatives and obtain higher order(directional or partial) derivatives:

Definition I.6 (Higher Order Derivatives). Let U ⊂ Rn be open and φ : U −→ R. If, forh1, . . . , hl ∈ Rn and i1, . . . , il ∈N0, the iterated directional derivatives(

∂h1

)i1(· · ·(

∂hk

)ilφ

),

exist, this is called a directional derivative of φ of order k = i1 + · · ·+ il . If the hj are taken to beelements of the standard basis of Rn, it is called a partial derivative of order k.

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We will also use the usual shorthand for higher order partial derivatives, i.e.,

∂kφ

∂ei1j1· · · ∂eil

jl

:=(

∂ej1

)i1(· · ·(

∂ejk

)ilφ

).

Definition I.7 (The Classes Ck). Let U ⊂ Rn be open and f = ( f1, . . . , fm) : U −→ Rm. If allpartial derivatives of orders up to k of all component functions f j exist and are continuous on U, wesay that f is k-times continuously differentiable on U and write f ∈ Ck(U; Rm). For k = ∞, wecall f smooth and write f ∈ C∞(U; Rm). (For m = 1, we shall simply write f ∈ Ck(U).)

We end this collection of definitions with some more vocabulary for specific types of maps.Let U ⊂ Rn and V ⊂ Rm be open and f : U −→ V. We say that f is

i) an immersion, if f is differentiable on U and D f∣∣x is injective for all x ∈ U,

ii) a submersion, if f is differentiable on U and D f∣∣x is surjective for all x ∈ U,

iii) an embedding, if f is a homeomorphism onto f (U) and an immersion,

iv) a (smooth) diffeomorphism, if f is a homeomorphism onto V and both f and f−1 are differ-entiable (smooth) on U and V, respectively.

Please observe that the dimensions n and m may pose obstructions to the existence of suchmaps: If n > m, there cannot be any immersions (and consequently embeddings), if m > n,there cannot be any submersions, and diffeomorphisms can only exist for n = m.

Remark I.8 (On Smoothness). It is often a matter of taste what exactly the word smooth refers to. Inthis course, by a smooth function we understand a function which is arbitrarily often continuouslydifferentiable; in other references, smoothness refers to a function being continuously differentiable asoften as one might need.

More importantly, when doing differential geometry, one can choose to consider smooth objectsonly: smooth submanifolds and surfaces, smooth functions between them, smooth vector fields etc.Or, one might be interested in studying the same questions for objects which are only of class Ck forsome finite k. Since this is an introductory course, we restrict ourselves to the study of smooth thingseven though most definitions and results carry over (almost) unchanged to the more general setting.Then again, we will need the smooth versions of the Inverse and Implicit Function Theorems, compareTheorems I.10 and I.12.

Important Theorems for Differentiable Maps

We close this review by stating (without proof) some important results: The first two concernthe differentials of compositions and inverses hence can be seen as algebraic properties of thedifferential (as a map on spaces of differentiable functions).

Theorem I.9 (Chain Rule). Let U ⊂ Rn1 , V ⊂ Rn2 be open subsets and f : U −→ Rn2 ,g : V −→ Rn3 . Let x ∈ U and suppose that f (U) ⊂ V.

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Then, if f is differentiable at x and g is differentiable at f (x), g f is differentiable at x and

D(g f )∣∣x = Dg

∣∣f (x) D f

∣∣x and J(g f )

∣∣x = Jg

∣∣f (x) · J f

∣∣x .

Theorem I.10 (Inverse Function Theorem). Let U ⊂ Rn be open, x ∈ U and f : U −→ Rn

be continuously differentiable (smooth).If D f

∣∣x is invertible, i.e., the Jacobian is nonsingular, then f is a local (smooth) diffeomorph-

ism, i.e., there are neighbourhoods U′ of x and V′ of f (x) such that f : U′ −→ V′ is a (smooth)diffeomorphism. In particular, f−1 : V′ −→ U′ is differentiable (smooth) and its differential aty = f (x) given by

D( f−1)∣∣

f (x) = D f∣∣−1x .

The next result is of a more computational value, though it is important for quite a few proofsas well. Depending on the source, it is named after Schwarz2 or Young3.

Theorem I.11 (Theorem of Schwarz-Young). Let U ⊂ Rn be open, x ∈ U, 1 ≤ i, j ≤ n and

f : U −→ R. If the partial derivatives ∂ f∂xi

, ∂ f∂xj

and ∂2 f∂xi∂xj

exist and are continuous on an open

neighbourhood of x, then the same holds for ∂2 f∂xj∂xi

and we have

∂2 f∂xi∂xj

=∂2 f

∂xj∂xi.

The last result shows in which circumstances a set of equations can be solved by use of animplicit function. Here, an implicit function is an auxiliary function having the set of solutionsto said equation as a level set, compare Section I.1.

Theorem I.12 (Implicit Function Theorem). Let U ⊂ Rn be open and f : U −→ Rm becontinuously differentiable (smooth). Suppose x0 ∈ Rn−m, y0 ∈ Rm so that (x0, y0) ∈ U. Letc = f (x0, y0).

If the matrix ∂( f1,..., fm)∂(y1,...,ym)

∣∣(x0,y0)

is invertible, then there are open neighbourhoods U′ ⊂ Rn−m

of x0 and U′′ ⊂ Rm of y0 and a continuously differentiable (smooth) map g : U′ −→ U′′ so that:For all x ∈ U′, y ∈ U′′ we have

f (x, y) = c ⇐⇒ y = g(x) .

2 Herrman Amandus Schwarz, ∗ 1843 in Hermsdorf, Silesia, † 1921 in Berlin3 Grace Chisholm Young, ∗ 1868 in Haslemere near London, † 1944 in England

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2016–17 · Differential Geometry Multivariate Integration · I.3

In this case, the differential of g is given by

∂(g1, . . . , gm)

∂(x′1, . . . , x′n−m)

∣∣x0

= −(

∂( f1, . . . , fm)

∂(y1, . . . , ym)

∣∣(x0,y0)

)−1 ∂( f1, . . . , fm)

∂(x′1, . . . , x′n−m)

∣∣(x0,y0)

.

Example I.13 (For the Implicit Function Theorem). Let n = 2, m = 1 and consider the function

f : R2 −→ R1 , f (x, y) = x2 + y2 .

For c = 1, the set

f−1(1) =(x, y) ∈ R2 ∣∣ x2 + y2 = 1

= S1

is the unit circle. Letting x0, y0 ∈ R, we see that

∂ f∂x∣∣(x0,y0)

= 2x0 ,∂ f∂y∣∣(x0,y0)

= 2y0 .

These “matrices” are invertible if and only if x0 6= 0 or y0 6= 0, respectively. If y0 6= 0, for instance, theImplicit Function Theorem tells us that we may implicitly solve the equation f (x, y) = 1 near (x0, y0)

for y, i.e., find a function g : U′ −→ U′′ satisfying

f (x, y) = 1 ⇐⇒ g(x) = y ,

near (x0, y0). Moreover, we obtain

∂g∂x∣∣x0

= −( ∂ f

∂y∣∣(x0,y0)

)−1· ∂ f

∂x∣∣(x0,y0)

= −(2y0)−1 · (2x0) = −

x0y0

.

It should be clear that, if for instance y > 0, g is given by g(x) =√

1− x2. This is also in line with theformula for the differential:

∂

∂x(1− x2)

12 = − x

(1− x2)12

= − xg(x)

= − xy

.

But please be aware that the Implicit Function Theorem does not give a construction for g, but onlyensures its existence!

I.3 Multivariate Integration

Let us now briefly sketch the theory of Lebesgue4 integrable functions. Again, we start with thebasic definitions and then give some important results.

Lebesgue integration

A cuboid in Rn is a cartesian product Q = I1 × · · · × In of nonempty, bounded intervals Ii ⊂ R.Its n-dimensional volume is defined as

voln(Q) := |I1| · · · |In| ,

4 Henri Lebesgue, ∗ 1875 in Beauvais, Oise, † 1941 in Paris

page 6


where |Ii| denotes the length of Ii. A step function is a finite linear combination of characteristicfunctions of cuboids,

ϕ(x) :=k

∑i=1

ci χQi (x) ,

and it is easy to define a sensible notion of integral for step functions:

∫Rn

ϕ(x) dx :=k

∑i=1

ci voln(Qi) (I.6)

Now, given any function f : Rn −→ R, where R = R∪ ±∞, we say that

Φ =∞

∑i=1

ci χQi

is an envelope for f , if the Qi are open cuboids, the coefficients are nonnegative, ci ≥ 0, and if| f (x)| ≤ Φ(x) for all x ∈ Rn. The content I(Φ) of an envelope is defined as the (infinite!) sum ofthe volumes of its cuboids, weighted by the appropriate coefficients:

I(Φ) :=∞

∑i=1

ci voln(Qi)

The notion of content allows us to define the L1–seminorm, which in turn leads to the space of(Lebesgue) integrable functions:

Definition I.14 (Lebesgue integral). Let f : Rn −→ R.

i) The L1–seminorm of f is defined as

‖ f ‖1 := inf

I(Φ)∣∣Φ is an envelope for f

.

ii) We say that f is (Lebesque) integrable, if there is a sequence ϕ1, ϕ2, . . . of step functions suchthat ‖ f − ϕk‖1 −→ 0 as k→ ∞. In this case, the (Lebesgue) integral of f is defined as∫

Rn

f (x) dx := limk→∞

∫Rn

ϕk(x) dx . (I.7)

iii) If U ⊂ Rn is open and f : U −→ R, we say that f is integrable on U if its extension f by 0to Rn is integrable and in this case write∫

U

f (x) dx :=∫

Rn

f (x) dx .

We denote the space of integrable functions by L1(U).

The space L1(U) is an infinite dimensional real vector space and the integral, as a mapping∫: L1 −→ R , (I.8)

page 7


is well-defined, linear and continuous with∣∣ ∫ f

∣∣ ≤ ‖ f ‖1. (Please note that it is not a boundedfunctional since we do not require the L1–seminorm to be finite.) Moreover, we have the follow-ing two important properties:

f ≤ g =⇒∫

f ≤∫

g

f ≥ 0 =⇒∫

f = ‖ f ‖1

But, what is probably most important: If a function R −→ R is Riemann integrable, then it isLebesgue integrable as well and the two integrals are the same. Thus, the actual computation ofLebesgue integrals of functions of one real variable is done in the usual way.

Important Theorems on Integrable Functions

With respect to the computation of integrals of multivariate functions, there are three importantresults: The Theorem of Fubini5 and the closely related Theorem of Tonelli6, and the theorem onintegration by substitution.

Theorem I.15 (Fubini). Let f ∈ L1(Rn) and write Rn = Rk ×Rl , x = (u, v) ∈ Rk ×Rl

where k + l = n. Then:

i) The function f ( · , v) : Rk −→ R, u 7−→ f (u, v) is integrable on Rk for almost everychoice of v ∈ Rl .

ii) The function F : Rl −→ R, v 7−→∫

Rk f (u, v) du is integrable on Rl .

iii) We have∫Rn

f (x) dx =∫Rl

( ∫Rk

f (u, v) du)

dv .

The Theorem of Fubini can be used to reduce a multivariate integral to multiple one dimen-sional integrals, with the sole prerequisite that f be integrable on Rn. What is often easier todecide is that f is locally integrable, i.e., for each point x ∈ Rn, there is a neighbourhood U ⊂ Rn

of x so that f∣∣U ∈ L

1(U). (In particular, any continuous function is locally integrable!) Then,the Theorem of Tonelli tells us, that we can use a Fubini-type calculation to decide whether f isintegrable on all of Rn:

Theorem I.16 (Tonelli). Let f : Rn −→ R be locally integrable. Then, f is integrable over Rn

if and only if there is a permutation π of 1, . . . , n so that the iterated integral∫R

· · ·∫R

∣∣ f (x1, . . . , xn)∣∣ dxπ(1) · · · dxπ(n)

5 Guido Fubini, ∗ 1879 in Venezia, † 1943 in New York City6 Leonida Tonelli, ∗ 1885 in Gallipolli, † 1946 in Pisa

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exists. In this case, we have∫Rn

f (x) dx =∫R

· · ·∫R

f (x1, . . . , xn) dxπ(1) · · · dxπ(n) ,

for any permutation π of 1, . . . , n.

The important ingredient to the calculation of integrals is the well-known theorem on integ-ration by substitution. This is a straight forward generalisation of the one dimensional case.

Theorem I.17 (Integration by Substitution). Let U, V ⊂ Rn be open, T : U −→ V be adiffeomorphism and f : V −→ R. Then, f is integrable over V if and only if ( f T)|det DT| isintegrable over U and, if so, we have∫

T(U)

f (y) dy =∫U

( f T)(x)∣∣det DT

∣∣x

∣∣ dx .

Example I.18 (Volume of Three-Balls). Let us, as an example, compute the volume of a ball

BR =

x ∈ R3 ∣∣ |x| ≤ R

in R3. This is, by definition, given by integrating the function 1 over BR:

vol3(BR) :=∫

BR

1 dx

Extending the function 1 to the cuboid [−R, R]3, and then using Theorem I.16, we see that 1 is integ-rable on [−R, R]3, but then it is integrable on BR ⊂ [−R, R]3 as well. Without going into the details ofnull sets (i.e. sets of measure 0), we note that∫

BR

f (x) dx =∫

BR\P

f (x) dx ,

where P is the half-plane given by P =

x ∈ R3∣∣ x1 = 0, x2 ≥ 0

. Now polar coordinates give a

diffeomorphism of BR \ P with a simple, open cuboid:

T : Q = (0, R)× (0, π)× (0, 2π) −→ BR \ P , (r, ϑ, ϕ) 7−→

r sin ϑ cos ϕ

r sin ϑ sin ϕ

r cos ϑ

Noting that |det DT| = |r2 sin ϑ| = r2 sin ϑ, substitution yields

vol3(B) =∫B

1 dx =∫

T(Q)

1 dx =∫Q

1 · r2 sin ϑ d(r, ϑ, ϕ) .

Then, applying Fubini’s Theorem twice, we get

vol3(BR) =∫

(0,R)

∫(0,π)

∫(0,2π)

r2 sin ϑ dϕdϑdr ,

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2016–17 · Differential Geometry Ordinary Differential Equations · I.4

and this evaluates to the well-known formula

vol3(BR) =43

πR3 .

I.4 Ordinary Differential Equations

Let us not delve to deeply into the rich theory of ordinary differential equations but only state adifferentiable version of the Theorem of Picard7-Lindelöf8 (which is sometimes also referred toas the Cauchy9-Lipschitz10 Theorem) and a result on the dependence on initial data.

An (ordinary) differential equation of order k is an equation of the form

x(k)(t) = F(t; x(t), x(1)(t), . . . , x(k−1)(t)

), (I.9)

where x(j) denotes the j-th derivative of x with respect to t and F : I ×Ω −→ Rn is continuous.Here, I ⊂ R and Ω ⊂ Rk are supposed to be open and connected sets. A (vector-valued)Ck–function x : I −→ Rn is a (global) solution to (I.9), if we have(

t, x(t), x(1)(t), . . . , x(k−1)(t))∈ I ×Ω (I.10)

for all t ∈ I, which is to say that x may be used as an input for the equation and

x(k)(t) = F(t; x(t), x(1)(t), . . . , x(k−1)(t)

), (I.11)

for all t ∈ I, which is to say that the vector(t, x(t), . . . , x(k)(t)

)indeed solves the equation.

An initial value problem for (I.9) consists of finding a solution to (I.9) which additionally satis-fies (

x, x(1), . . . , x(k−1))(t0) :=(

x(t0), x(1)(t0), . . . , x(k−1)(t0))= y0 , (I.12)

where (t0, y0) ∈ I ×Ω is given.

Important Theorems for ODEs

There are two results which are of interest to us: Most importantly the existence and uniquenessof solutions to initial value problems and then a result which shows that if we perturb the initialvalue problem slightly (in a differentiable manner), then the solution is perturbed only slightly(and differentiably) as well:

7 Émile Picard, ∗ 1856 in Paris, † 1941 ibidem8 Ernst Leonard Lindelöf, ∗ 1870 in Helsinki, † 1946 ibidem9 Augustin-Louis Cauchy, ∗ 1789 in Paris, † 1857 in Sceaux10 Rudolf Lipschitz, ∗ 1832 in Königsberg (Kaliningrad), † 1903 in Bonn

page 10

2016–17 · Differential Geometry Ordinary Differential Equations · I.4

Theorem I.19 (of Picard-Lindelöf). Let I ⊂ R and Ω ⊂ Rk be open and connected sets andsuppose that F : I ×Ω −→ Rn is continuous in (t, y) ∈ I ×Ω and differentiable in y ∈ Ω. Let(t0, y0) ∈ I ×Ω. Then, there is a solution x : I −→ Rn to the initial value problem

x(k)(t) = F(t; x(t), . . . , x(k− 1)(t)

),

(x, . . . , x(k−1))(t0) = y0

and this solution is uniquely determined by F and (t0, y0).

Proposition I.20 (Dependence on Parameters). In the situation of Theorem I.19, assumethat the data F and (t0, y0) depend differentiably on a parameter λ. Then, the solution givenby Theorem I.19 depends differentiably on λ as well.

page 11

References and Further Reading

The following list just aims at giving a first place to look for further details, additional or differentexplanations or more references. There are far too many good text books on analysis, generaltopology and differential geometry to name them all. The book being closest to what we aredoing here is [dC1], although the notation might differ to some extent, followed by [Pr]. Despitethe age of its first edition, [Sp1] is still a very good textbook on differential geometry, though theamount of five volumes is a bit intimidating.

On Differential and Riemannian Geometry:

[dC1] Manfredo P. do Carmo, Differential Geometry of Curves and Surfaces, Prentice-Hall, Inc.,Englewood Cliffs, NJ, 1976.

[dC2] Manfredo P. do Carmo, Riemannian Geometry, 2nd. pr., Birkhäuser, Basel, 1992.

[Pr] Andrew Pressley, Elementary Differential Geometry, Springer, London, 2001.

[Sp1] Michael Spivak, A Comprehensive Introduction to Differential Geometry : Volumes I–V, 3rded., Publish or Perish, Houston, TX, 1999.

On Analysis and General Topology:

[Ar] V. I. Arnold, Ordinary Differential Equations, 2nd pr., Springer, Berlin – Heidelberg, 2006.

[PM] M. H. Protter and C. B. Morrey, A First Course in Real Analysis, Springer, London, 1991.

[Ru] Walter Rudin, Principles of Mathematical Analysis, 3rd ed., McGraw-Hill, New York, NY,1976.

[Sp2] Michael Spivak, Calculus on Manifolds, Perseus Books Publishing, New York, NY, 1965.

page 12

Introduction

What is “Differential Geometry”? Classically, geometry is the study of shapes and bodies as forexample triangles, (regular) polygons, circles and tetrahedrons, platonic bodies, spheres,... Youwill also know the geometry of linear or affine spaces from linear algebra: For instance, the linearstructure of the plane R2 and three dimensional space R3, how to add and subtract vectors,define lengths of and angles between vectors etc. The advent of calculus in the 17th centuryallowed for a systematic and rigorous treatment of curved shapes as well, most importantlycurves and surfaces in R2 and R3. This is where the term “Differential” is being added to“Geometry”. Moreover, there are two pairs of view points in Differential Geometry: local vs.global and extrinsic vs. intrinsic. Local properties only see how shapes look like in arbitrarilysmall patches (when you look at a few square metres of calm sea, it seems to be flat) whileglobal properties take into account all of the curve or surface (Earth, as a whole, does not appearto be flat at all). The other pair distinguishes between properties that depend on how we embedthe object in an ambient space (extrinsic) and those which do not depend on any ambient space(intrinsic). One example here is the fact that cartographers have known for centuries that thereis no plane map of any part of the surface of Earth which does not distort lengths, i.e., in whichall lengths are in proportion. This points to an intrinsic property of the sphere (Earth) whichdoes not depend on how we realise it (as a map).

Briefly, “Differential Geometry” is the study of the geometry of curves and surfaces (or gener-alisations thereof) using tools from calculus and (linear) algebra, linking local and global quant-ities and properties, while being aware of the use of extrinsic information.

This course intends to introduce you to the geometry of curves and surfaces. Not only is thisthe historical starting point for this field in mathematics, also, most of the concepts introducedhere generalise to higher dimensions. But the study of these generalisations requires more toolsand is the subject of different courses, for instance on algebraic varieties in “Algebraic Geometry”or on Riemannian manifolds in “Riemannian Geometry”. We will also focus on differentiable orsmooth objects only, and not delve into what is called “Differential Topology”, and not considerobjects which have a boundary or more complicated singularities. The geometry, topology andanalysis of such “singular spaces” is a field of ongoing research in mathematics.

We start this course with the study of curves, giving a complete local description of curves inR3 and a global result on plane curves. In Chapters III and IV, we introduce surfaces and theircurvatures. Then, in Chapter V, we prove Gauss’ famous Theorema Egregium (showing that atleast a certain part of the curvature of a surface is intrinsic) and give a complete local descriptionof surfaces in R3. Finally, Chapters VI and VII see the combination of what we will have done sofar: On the one hand the treatment of curves on surfaces (in particular the important geodesics)and on the other hand linking local information (curvature) to global information (the Eulercharacteristic) in the celebrated Theorem of Gauss-Bonnet.

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Chapter II

The Theory of Curves

In this chapter, we define (parameterised) curves as functions γ : I −→ Rn and, for n = 3,study their local properties. A key instrument is the Frenet11-Serret12 frame, an orthonormalmoving frame reflecting the geometry of the curve. This frame naturally leads to the definitionof curvature and torsion and we will show that these two scalar functions essentially determinethe curve. Then, we will focus on curves in R2 and study global properties of simple closedplane curves, generalising a classical result on the sum of angles in a polygon.

II.1 Parameterised Curves

Parameterisations and Reparameterisations

Definition II.1 (Parameterised Curve). A parameterised curve in Rn is a continuously differ-entiable map γ : I −→ Rn, where I ⊂ R is an intervall. Moreover, we define the trace of γ to be theset tr(γ) := γ(I) ⊂ Rn and say that γ is regular if γ′(t) := dγ(t)

dt 6= 0 for all t ∈ I. The vector γ′(t)is called the tangent vector to γ at γ(t).

Example II.2.i) γ(t) = (a cos t, a sin t, bt), t ∈ R, parameterises a helix. Since

γ′(t) = (−a sin t, a cos t, b) 6= 0 for all t ∈ R ⇐⇒ a 6= 0 or b 6= 0 ,

this is a regular curve if and only if a2 + b2 6= 0.

ii) γ(t) = (t3, t2), t ∈ R, is not a regular curve: Since γ′(t) = (3t2, 2t), we have γ′(0) = 0.

Remark II.3. One can similarly define parameterised curves of class Ck for any k ∈ N0 or smooth curves of

class C∞. Please note that the tangent vector will be defined for k > 0 only. The parameter t can be thought of as being time and tr(γ) the trajectory of a particle. In this

case, γ′(t) corresponds to the instantaneous velocity of said particle at time t.

We can clearly think of two parameterised curves γ1, γ2 having the same trace but neverthe-

11 Jean Frédéric Frenet, ∗ 1816 in Périguex, France, † 1900 ibidem.12 Joseph Alfred Serret, ∗ 1819 in Paris, † 1885 in Versailles

page 14

2016–17 · Differential Geometry Parameterised Curves · II.1

less being different. For instance, think of two particles moving on the same trajectory but atdifferent speeds or in different directions. To formalise this for parameterised curves, we needto introduce homeo– and diffeomorphisms.

Definition II.4 (Homeo– and Diffeomorphisms). Let I, J ⊂ R be open intervals and supposeφ : I −→ J is a bijection. φ is called a

i) homeomorphism if both φ and φ−1 are continuous,

ii) diffeomorphism if both φ and φ−1 are continuously differentiable.

Just like linear maps preserve the linear structure of vector spaces, homeomorphisms preservethe topological structure. For instance, if φ : R −→ R is a homeomorphism, then I ⊂ R is open ifand only if φ(I) is open. Diffeomorphism take this one step further by providing a differentiabletransformation of R. Using the Inverse Function Theorem, Theorem I.10, it is easy to see thata bijection φ : I −→ J is a diffeomorphism, if and only if φ is continuously differentiable andφ′(t) 6= 0 for all t ∈ I.

Remark II.5 (Diffeomorphisms of Closed Intervals). Diffeomorphisms are defined on open intervals be-cause we need to be able to talk about differentiability. But we can consider diffeomorphisms ofclosed intervals as well, something that we might need to do later: If I ⊂ R is a closed interval andφ : I −→ J ⊂ R is a bijection, we call φ a diffeomorphism, if we can extend it to be a diffeomorph-ism: If there are open intervals I, J containing I and J, respectively, and if there is a diffeomorphismφ : I −→ J such that φ(t) = φ(t) for all t ∈ I.

Definition II.6 (Reparameterisation). Let γ1 : I −→ Rn and γ2 : J −→ Rn be parameterisedcurves. If there is a diffeomorphism φ : I −→ J satisfying γ2 = γ1 φ, we call γ2 a reparameterisa-tion of γ1.

A curve in Rn is an equivalence class of parameterised curves, where two curves are called equi-valent if and only if one is a reparameterisation of the other.

Note that the definition of curves as equivalence classes is insofar incomplete as we needto show that “there is a reparameterisation linking the curves” defines an equivalence relation.This is left as an exercise.

Example II.7. The parameterised curves γ1(t) = (cos t, sin t, t) and γ2(t) = (cos 2t, sin 2t, 2t), t ∈ R, arehelices. For φ(t) = 2t, which is a diffeomorphism, we have γ2 = γ1 φ hence they are equivalent andhave the same traces: tr(γ1) = tr(γ2). The tangent vectors depend on the parameterisation, though:

γ′1(t) = (− sin t, cos t, 1) whereas γ′2(t) = (−2 sin t, 2 cos t, 2)

page 15


Remark II.8.i) Regularity does not depend on the parameterisation and therefore is a property of curves (as

equivalence classes): Since ddt (γ1 φ)(t) = γ′1

∣∣φ(t) · φ

′(t) and φ′(t) 6= 0, we have

|γ′1(t)| 6= 0 for all t ∈ I ⇐⇒ |γ′2(t)| 6= 0 for all t ∈ J .

ii) Usually, when using the term “curve”, we will mean a parameterised curve, as it is the explicitparameterisation which enables us to make computations. Whenever we want to refer to curvesas equivalence classes, this will be stated explicitly.

Arc Length

It is very often useful and convenient to consider a certain special parameterisation of a curve.In the moving particle picture, this parameterisation can be thought of as being the unit speedparameterisation. The advantage of using this special parameterisation will become clear in thefollowing sections.

Definition II.9 (Arc Length). The arc length of a parameterised curve γ : I −→ Rn is

`(γ) :=∫I

|γ′(t)|dt .

If we have

`(γ|[t1,t2]) = t2 − t1 for all t1, t2 ∈ I,

we say that γ is parameterised by arc length or of unit speed.

Intuitively, Definition II.9 just says that a curve is parameterised by arc length, if and only ifthe length of each segment γ([t1, t2]) of tr(γ) is the same as the length of [t1, t2]. The interpret-ation of this as a unit speed curve comes from the next proposition. We will usually denote thearc length parameter by s, i.e., write γ(s) if γ is parameterised by arc length. Derivatives withrespect to this parameter will be denoted by a dot, e.g. γ(s).

Proposition II.10.

i) Arc length does not depend on the parameterisation of the curve.

ii) γ is parameterised by arc length if and only if |γ′(t)| = 1 for all t ∈ I.

iii) Any regular curve can be parameterised by arc length.

Proof :i) This is essentially integration by substitution: Suppose I = [a, b] and let φ : I −→ J be a

page 16


diffeomorphism such that φ(a) < φ(b). For γ2 = γ1 φ we then have

`(γ2) =

b∫a

|γ′2(t)|dt =b∫

a

| ddt (γ1 φ)(t)|dt =

b∫a

|γ′1(φ(t)) · φ′(t)|dt

=

φ(b)∫φ(a)

|γ′1(r)|dr = `(γ1) ,

where we have used φ′(t) > 0 in the second to last step. If φ(a) > φ(b), φ′(t) is negativeand this negative sign is cancelled out by the reversal of orientation in integration.

ii) If |γ′(t)| = 1 for all t ∈ I, then clearly `(γ|[t1,t2]) =

∫ t2t1|γ′(t)|dt = t2 − t1. On the other

hand, if `(γ|[t1,t2]) = t2 − t1 for all t1, t2 ∈ I, then differentiation with respect to t gives

|γ′(t)| = ddt

t∫t1

|γ′(r)|dr =ddt(t− t1) = 1 .

iii) Let γ : I −→ Rn be regular, i.e., γ′(t) 6= 0 for all t ∈ I. Suppose again that I = [a, b]and let ψ(t) :=

∫ ta |γ

′(r)|dr. Since ψ′(t) = |γ′(t)| 6= 0 for all t ∈ I, ψ : I −→ [0, `(γ)] is adiffeomorphism and so is ψ−1. Consider the parameterised curve γ(s) = γ(ψ−1(s)) andfix t1, t2 ∈ I. Letting s1 = ψ(t1), s2 = ψ(t2) and assuming for the moment that s1 < s2, weget

s2 − s1 = ψ(t2)− ψ(t1) = `(γ|[a,t2])− `(γ|[a,t1]

) = `(γ|[t1,t2]) = `(γ|[s1,s2]

) ,

where the last equality follows from item i). Thus, γ is a reparameterisation of γ by arclength. (If s1 > s2, the negative sign is again cancelled by the reversal of orientation inintegration.)

Example II.11.i) Consider the helix from Example II.2, item i) again, γ(t) =

(a cos t, a sin t, bt

), where t ∈ R and

a2 + b2 6= 0. Then,

|γ′(t)| =∣∣(− a sin t, a cos t, b

)∣∣ = √a2 + b2

is independent of t. Letting t = s√a2+b2 , we therefore obtain

|γ′(s)| = 1√a2+b2

√a2 + b2 = 1 .

Thus, γ(s) =(a cos s√

a2+b2 , a sin s√a2+b2 , bs√

a2+b2

)is a reparameterisation by arc length.

ii) Let γ(t) = (cos2 t, sin2 t, sin2 t + 1). Using sin t cos t = 12 sin 2t, we get

γ′(t) = (−2 cos t sin t, 2 cos t sin t, 2 cos t sin t) = sin 2t · (−1, 1, 1) ,

and since this vanishes exactly if t ∈ π2 Z, γ is regular on any interval I such that I ∩ π

2 Z = ∅.Moreover,∣∣γ′(t)∣∣ = √3

∣∣ sin 2t∣∣ .

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2016–17 · Differential Geometry The Local Theory of Space Curves · II.2

In order to obtain a reparameterisation for γ by arc length on the interval, say, I = (0, π2 ),

consider (as was done in the proof of item iii) of Proposition II.10) the function

ψ(t) =t∫

0

|γ′(r)|dr =√

3t∫

0

sin 2rdr = −√

32

cos 2r∣∣∣t0=

√3

2(1− cos 2t

)=: s(t) .

Its inverse is given by t(s) = ψ−1(s) = 12 arccos

(1 − 2√

3s)

and using this as a reparameter-

isation, we set γ(s) = γ(t(s)

). Then, because we have identities d

ds arccos s = − 1√1−s2 and

sin arccos s =√

1− s2,∣∣γ′(s)∣∣ = ∣∣γ′(t(s))t′(s)∣∣=√

3∣∣∣ sin arccos(1− 2√

3s)∣∣∣ · 1√

3

1√1− (1− 2√

3s)2

=√

3

√1−

(1− 2√

3s)2

1√3

1√1− (1− 2√

3s)2

= 1 .

Thus, we have found a parameterisation of γ by arc length. (Observe that ψ′(t) =√

3 sin 2t = 0precisely for t ∈ π

2 Z. So, when restricted to intervals I as above, this is indeed a diffeomorph-ism.)

As mentioned in item ii) of Example II.11, the proof of item iii) of Proposition II.10 gives arecipe for the calculation of a parameterisation by arc length: Compute the integral s(t) = ψ(t) =∫ t

a |γ′(r)|dr, determine its inverse t(s) := ψ−1(s) and use it to reparameterise: γ(s) = γ

(t(s)

)will be parameterised by arc length. But keep the following in mind:

This recipe will only work if γ is regular. For if, say, γ′(t0) = 0, then ψ′(t0) = 0 and ψ willnot be a diffeomorphism. Even if γ is regular, it might be that we cannot use this recipe to obtain an explicit repara-

meterisation since we have to explicitly compute the integral ψ(t) =∫ t

t0|γ′(r)|dr. Even for

seemingly simple curves like ellipses in R2, this is not possible using elementary functionsonly. This is an old problem in mathematics and leads to the theory of elliptic integrals.

II.2 The Local Theory of Space Curves

In this section, we will focus solely on curves in R3. By Proposition II.10 we know that we canparameterise any regular curve by arc length and for the rest of this section we will assume thatany curves are regular and parameterised by arc length. Moreover, as will become clear in amoment, we will assume that all curves are at least twice continuously differentiable.

The Frenet-Serret Apparatus

In this case, the tangent vector γ is of length 1 and we will denote this unit tangent vector byT(s) := γ(s). Since

⟨T(s) , T(s)

⟩= |T(s)|2 = 1, differentiating both sides with respect to s yields

0 =dds

1 =dds⟨

T(s) , T(s)⟩=⟨

T(s) , T(s)⟩+⟨

T(s) , T(s)⟩= 2

⟨T(s) , T(s)

⟩. (II.1)

page 18


Therefore, the vector T(s) is orthogonal to T(s): T(s) ⊥ T(s) for all s.

Remark II.12. Just in case you haven’t come across this situation before, we will briefly show how todifferentiate a term like

⟨T(s) , T(s)

⟩with respect to s. The first step is to consider this as a function

of two variables, f (s, t) =⟨

T(s) , T(t)⟩. Letting g(s) = (s, s), we have

⟨T(s) , T(s)

⟩=(

f g)(s) and its

derivative with respect to s is given by

dds⟨

T(s) , T(s)⟩=

dds(

f g)(s) = D f

∣∣g(s) · Dg

∣∣s ,

using the chain rule, Theorem I.9. Now since

D f∣∣(s,t) =

(dds f (s, t), d

dt f (s, t))=(⟨

T′(s) , T(t)⟩,⟨

T(s) , T′(t)⟩)

and

Dg∣∣s =

(11

),

we obtain

D f∣∣g(s) · Dg

∣∣s =

(⟨T′(s) , T(t)

⟩,⟨

T(s) , T′(t)⟩) ∣∣∣

t=s·(

11

)=⟨

T′(s) , T(t)⟩+⟨

T(s) , T′(t)⟩∣∣∣

t=s= 2

⟨T(s) , T′(s)

⟩.

Definition II.13 (Frenet-Serret Frame). Let γ : I −→ R3 be a regular C2–curve, parameterisedby arc length. The scalar function

κ(s) := |T(s)|

is called the curvature of γ at γ(s). If κ(s) 6= 0, we further define

N(s) :=1

κ(s)T(s) =

T(s)|T(s)|

and B(s) := T(s)× N(s) .

N(s) is called the principal unit normal and B(s) the binormal to γ. The resulting right-handedmoving frame T, N, B is called the Frenet-Serret frame of γ.

Remark II.14 (Frames, Right-Handedness and the Cross-Product).i) By a frame, we mean an ordered basis, in this case of R3. In contrast to a basis, a frame hence

always determines an orientation. A moving frame on I ⊂ R is a continuous function on I whichgives a frame for every s ∈ I.

ii) Let e1, e2, e3 be the standard basis for R3. Given another basis v1, v2, v3, there is a uniquelinear map L : R3 −→ R3 such that L(ei) = vi for i = 1, 2, 3. This map is nonsingular and willthus have either positive or negative determinant. If det L > 0, we say that the frame v1, v2, v3is right-handed, otherwise we call it left-handed. (This of course depends on the ordering of thebasis e1, e2, e3, i.e., on the choice of orientation for R3.)

page 19


iii) Given two linearly independent vectors u, v ∈ R3, there is a unique vector w satisfying thefollowing three properties: i) w is perpendicular to both u and v, ii) the ordered set u, v, w isa right-handed frame of R3 and iii) |w| equals the area of the parallelogram generated by u andv. This vector w is called the cross product of u and v. In particular,

w = u× v =((

u2v3 − u3v2),−(u1v3 − u3v1

),(u1v2 − u2v1

)),

where u = (u1, u2, u3), v = (v1, v2, v3). The cross product is bilinear and skew-symmetric,where the latter means that u × v = −v × u. (Similarly, the cross product can be defined forn− 1 vectors in Rn, i.e., as a multi-linear and alternating map

(Rn)n−1−→ Rn. But we will not

make use of this generalisation.)

Please note that by differentiating the defining equality B = T × N and using

T = κ N , (II.2)

we see that

B = T × N + T × N = κN × N + T × N = T × N . (II.3)

Hence B is orthogonal to T and since B is of unit length, the same calculation we did for T andT shows that B is orthogonal to B as well. Consequently there is a scalar function τ(s) such thatB(s) = τ(s)N(s).

Definition II.15 (Torsion). Let γ : I −→ R3 be C2, regular and parameterised by arc length. Thescalar function τ : I −→ R, which is given by B(s) = τ(s)N(s) whenever s ∈ I satisfies κ(s) 6= 0,and by τ(s) = 0 else, is called the torsion of γ.

In (II.2) and (II.3), we have already seen formulae for T and B. To complete the presentation,note that orthonormality and right-handedness of the Frenet-Serret frame T, N, B imply N =

B× T and consequently

N = B× T + B× T = τN × T + κB× N = −τB− κT . (II.4)

Definition II.16 (Frenet-Serret Formulae). The equations

T(s) = κ(s)N(s) (II.5)

N(s) = −κ(s)T(s)− τ(s)B(s) (II.6)

B(s) = τ(s)N(s) (II.7)

are called the Frenet-Serret formulae. They can be written in matrix form as

F(s) = A(s)F(s) , (II.8)

where

F =

TNB

and A =

0 κ 0−κ 0 −τ

0 τ 0

.

page 20


Please note that, for any s ∈ I, A(s) is a skew-symmetric 3-by-3 matrix and that, wheneverκ(s) 6= 0, F(s) ∈ SO(3), the group of orthogonal (Mt M = MMt = id) 3-by-3 matrices of determ-inant 1. The collection T, N, B, κ, τ is sometimes referred to as the Frenet-Serret apparatus.

Example II.17. Once again, we consider the helix

γ(s) =(

a cos s√a2+b2 , a sin s√

a2+b2 , bs√a2+b2

),

where s ∈ R and a2 + b2 6= 0. We will compute the full Frenet-Serret apparatus of γ. In Example II.11,we have seen that γ is parameterised by arc length and that

T(s) =1√

a2 + b2

(−a sin s√

a2+b2 , a cos s√a2+b2 , b

).

Then,

T(s) =1

a2 + b2

(−a cos s√

a2+b2 ,−a sin s√a2+b2 , 0

)and the curvature is consequently given by κ(s) = a

a2+b2 . The principal unit normal vector is

N(s) =T(s)κ(s)

= −(

cos s√a2+b2 , sin s√

a2+b2 , 0)

and the unit binormal is

B(s) = T(s)× N(s) =1√

a2 + b2

(b sin s√

a2+b2 ,−b cos s√a2+b2 , a

).

Finally,

B(s) =b

a2 + b2

(cos s√

a2+b2 , sin s√a2+b2 , 0

)= − b

a2 + b2 N(s) ,

which shows that τ(s) = − ba2+b2 .

Remark II.18.i) The full Frenet-Serret frame T, N, B is defined only if κ(s) 6= 0, as we need to normalise T(s)

and κ(s) =∣∣T(s)∣∣. This makes sense for another reason: N, B is a distinguished frame for the

plane P(s) which is orthogonal to T(s), the distinguishing factor being that T(s) turns in thedirection of N(s) as we move along the curve. If κ(s) = 0, T(s) does not vary and we have noway of canonically picking a frame for P(s).

ii) κ(s) quantifies how T(s) turns in the direction of N(s), i.e., measures curvature in the T-N-plane. τ(s) on the other hand measures curvature in the N-B-plane. Compare this to the valuesof κ and τ in Example II.17, item ii), for a = 0 or b = 0.

The Fundamental Theorem of the Local Theory of Space Curves

Remember that we are only dealing with curves in R3 at the moment. Having the Frenet-Serret apparatus and formulae at hand, we can now show that curvature and torsion essentiallydetermine any regular curve. We start by considering the role of torsion.

page 21


Proposition II.19. Let γ : I −→ R3 be a regular C2-curve, parameterised by arc length. Then,the torsion τ(s) of γ vanishes identically if and only if tr(γ) is contained in a plane, i.e., in a2-dimensional affine subspace of R3.

Proof : If tr(γ) is contained in a plane P, then T(s) and N(s) need to be parallel to P for all s.So, if ν denotes a unit normal of P, either B(s) = ν or B(s) = −ν for all s. In any case, B(s) = 0and so τ(s) = 0 for all s.

Now assume τ(s) = 0 for all s. Then, B(s) = τ(s)N(s) = 0 and hence B(s) = B0 for all s. Forany s0, we then obtain

dds(⟨

γ(s)− γ(s0) , B0⟩)

=⟨

T(s) , B0⟩+ 0 = 0

and therefore⟨γ(s)− γ(s0) , B0

⟩= c ∈ R for all s. Inserting s = s0, we see that c = 0. Hence,

there are vectors v = γ(s0) and w = B0 such that for all s,

γ(s) ∈

x ∈ R3 ∣∣ x− v ⊥ w

,

which is a plane.

Proposition II.19 shows that the torsion measures the extend to which a curve is not a planecurve. Given the additional datum of curvature, this determines the curve up to Euclidean13

motion. A Euclidean or rigid motion on Rn is a map of the form

T : Rn −→ Rn , x 7−→ T(x) := ρ · x + v , (II.9)

where ρ ∈ SO(n) is a rotation and v ∈ Rn. It is intuitively clear that curvature and torsion shouldnot capture the “positioning” of the curve in space and hence are invariant under rotations andtranslations.

Theorem II.20 (Fundamental Theorem of Curves). Let I ⊂ R be an open interval and sup-pose there are continuous functions κ, τ : I −→ R with κ(s) 6= 0 for all s ∈ I.

i) There is a regular curve γ : I −→ R, parameterised by arc length, whose curvature andtorsion are given by κ and τ. The curve γ is unique up to Euclidean motions.

ii) Given s0 ∈ I and γ0, T0, N0 ∈ R3 so that T0 ⊥ N0, there is a unique curve γ : I −→ R3,which is parameterised by arc length, has curvature and torsion given by κ and τ andsatisfies the initial conditions γ(s0) = γ0, T(s0) = T0 and N(s0) = N0.

Proof : We will proceed as follows: Given any right-handed orthonormal frame T0, N0, B0 andany point s0 ∈ I, we show that these initial conditions and the data κ and τ give rise to a

13 Euclid of Alexandria, est. mid 4th – mid 3

rd century BC

page 22


moving frame T(s), N(s), B(s) which is orthonormal, right-handed and satisfies the Frenet-Serret formulae for any s ∈ I. Then, we will construct a curve γ whose Frenet-Serret frameexactly is T, N, B.

Let T0, N0, B0 be an orthonormal and right-handed frame and s0 ∈ I. The system of equa-tions (II.5) – (II.7) (or equivalently (II.8)) is a linear system of nine ordinary differential equationsand Theorem I.19 shows that there is a unique solution T, N, B satisfying the nine initial con-ditions

T(s0) = T0 , N(s0) = N0 , B(s0) = B0 .

In order to show that this is a right-handed orthonormal frame, consider the matrix-valuedfunction M(s) = F(s)F(s)t with F(s) defined by T, N and B as in Definition II.16. Then,

M(s) = F(s)F(s)t + F(s)(

F(s)t ) = A(s)M(s)−M(s)A(s) (II.10)

and since T0, N0, B0 is a right-handed orthonormal frame we see that

M(s0) = id . (II.11)

Next, observe that the identity matrix id(s) = id clearly solves the system (II.10) with initialconditions (II.11). By the uniqueness part of Theorem I.19, we see that M(s) = id for all s ∈ I.But M(s) being the identity matrix for all s ∈ I is equivalent to F(s) being orthonormal for alls ∈ I. Finally, we note that det F(s) is a continuous function of s ∈ I and det F(s) = ±1 forall s ∈ I, det F(s0) = 1. The latter holds since T0, N0, B0 was assumed to be right-handed.Then, det F(s) = 1 for all s ∈ I and we have shown that T, N, B is indeed a right-handedorthonormal frame for all s ∈ I.

As a next step, we define a curve γ : I −→ R3 by

γ(s) =s∫

s0

T(r)dr ,

where s0 ∈ I and the integral is taken component-by-component. The Fundamental Theoremof Calculus shows that T is the unit tangent vector of γ and since T and N solve (II.5), N is γ’sprincipal unit normal and κ its curvature. Moreover, due to the right-handedness of the frameT, N, B, we have B = T × N, which shows that B is the binormal of γ and by equation (II.7),τ is its torsion. Hence we have shown the existence of a curve γ : I −→ R3 with curvature andtorsion given by κ respectively τ.

Let us now address uniqueness. Let γ : I −→ R3 be a second curve with curvature κ andtorsion τ, we denote the matrix corresponding to its Frenet-Serret frame by F(s). Since any tworight-handed orthonormal frames are related by a rotation, there is ρ ∈ SO(3) such that

F(s0) = (ρ F)(s0) .

Now we define a third matrix-valued function by

F(s) = (ρ−1 F)(s) .

page 23

2016–17 · Differential Geometry The Global Theory of Plane Curves · II.3

We want to show that in fact F(s) = F(s) for all s ∈ I, i.e., that the frames F(s) and F(s) arerelated by the same rotation ρ for all s ∈ I. First of all,

dds∥∥F(s)− F(s)

∥∥22 =

dds

(∣∣T − T∣∣2 + ∣∣N − N

∣∣2 + ∣∣B− B∣∣2)

= 2(⟨

T − T , ˙T − T⟩+⟨

N − N , ˙N − N⟩+⟨

B− B , ˙B− B⟩)

= 0 ,

using equations (II.5) – (II.7). Thus, ‖F(s) − F(s)‖2 = c is independent of s and since clearlyF(s0) = F(s0), we see that F(s) = F(s) for all s ∈ I.

But then, T(s) = ρT(s) and consequently ˙γ(s) = ργ(s) for all s ∈ I. Integration with respectto s now shows that there is a constant vector v ∈ R3 such that γ = ργ + v, i.e., γ and γ differby a Euclidean motion.

With regard to the second item, observe that the choice of γ0 fixes the choice of translation v,whereas the choice of T0 and N0 fixes the choice of rotation ρ. Using the first part, these choicesuniquely determine the curve.

Remark II.21. If γ is a regular curve which is not of unit speed, we can nevertheless define its curvature:Let γ(s) = γ

(t(s)

)be a reparameterisation by arc length and simply define κ(t) = κ

(s(t)

). As there

are only two reparameterisations by arc length (corresponding to opposite orientations), it is easy tosee that κ(t) does not depend on the choice of reparameterisation. If κ(t) 6= 0, we can similarly definethe full Frenet-Serret apparatus T(t), N(t), B(t) and τ(t). But be aware that you will still need theunit speed parameterisation to actually compute the apparatus and that the Frenet-Serret formulaehold for the arc length parameter s only!

II.3 The Global Theory of Plane Curves

As any curve in R2 can easily be considered a curve in R3 as well, the local results from thelast section carry over to regular plane curves. But the first truly global results we considerhold for plane curves only. To begin with, we mention, without giving proofs, three almostclassical results on simple plane curves: Jordan’s14 Curve Theorem, the Isoperimetric Inequalityand Hopf’s15 Umlaufsatz (or the Theorem of Turning Tangents). Then, we generalise the latterto curvilinear polygons.

Definition II.22 (Simple and Closed Curves). Let γ : [a, b] −→ Rn be a parameterised curve.We say that γ is simple, if γ

∣∣[a,b) is injective and call it closed if γ(a) = γ(b). A simple closed plane

curve is often called a Jordan curve.

Thus, simple closed curves are what we intuitively think of when we talk about loops. They arean essential tool in studying the topology of surfaces (or more general spaces).

14 Marie Ennemond Camille Jordan, ∗ 1838 in Lyon, † 1922 in Paris15 Heinz Hopf, ∗ 1894 near Wrocław, † 1971 in Zollikon, Switzerland

page 24


Three Classical Theorems

We will not prove the following classical results, but use them as black boxes for what is to come.Proofs can be found in [dC1], for instance.

• It is intuitively clear though not trivial to prove that a simple closed plane curve cuts R2

into two pieces: One unbounded, exterior part and one bounded, interior part:

Theorem II.23 (Jordan’s Curve Theorem). Let γ be a simple closed plane curve. Then, theset R2 \ tr(γ) is the disjoint union of two open sets, exactly one of which is bounded.

Definition II.24. The bounded open set from Theorem II.23 is called the interior of the curve γ anddenoted by int(γ).

If γ is regular and parameterised in such a way that int(γ) lies on the left side of the oriented lineγ(t) + λγ′(t)

∣∣ λ ∈ R

, for any t ∈ I, we say that γ is parameterised or oriented positively.

As we have stated, Theorem II.23 holds for any continuous simple closed curve, not only fordifferentiable ones. It is in fact a homological result and the most elegant and shortest proofsrely on this theory.

• The second theorem we want to mention, the Isoperimetric Inequality, is a result onthe area of int(γ). It has been long known, probably already in Ancient Greece, that the areaenclosed by a loop of thread is always less or equal to the area of a disc with the circumferencegiven by the length of the thread. The Isoperimetric Inequality now states that, fixing the arclength, the circle maximises the area of the interior.

Theorem II.25 (The Isoperimetric Inequality). Let γ be a simple closed plane curve of arclength `(γ). Then,

4π area(int(γ)

)≤ `(γ)2 , (II.12)

and equality holds if and only if γ parameterises a circle of radius 12π `.

More generally, Theorem II.25 holds for any simple closed curve for which we can sensiblydefine arc length and area of the interior. Assuming that a maximiser of area exists, the first proofwas given by Steiner16, a full proof, including the existence of a maximiser, was given first byWeierstrass17 using his newly developed calculus of variations.

• Closely related to Jordan’s Curve Theorem is the Theorem of Turning Tangents or Hopf’sUmlaufsatz (German for winding theorem). It quantifies the total variation of the tangent vector

16 Jakob Steiner, ∗ 1796 in Utzenstorf, Switzerland, † 1863 in Bern17 Karl Theodor Wilhelm Weierstrass, ∗ 1815 in Ostenfelde, Prussia, † 1897 in Berlin

page 25


γ′(t) as we move along the curve. But more importantly, it does so in a very general way, whichwe will use at the end of the course to proof the Theorem of Gauss-Bonnet.

In the following, by an orthonormal moving frame along γ : I −→ R2, we understand anordered pair V, W of vector-valued, continuously differentiable functions so that, for eacht ∈ I, V(t), W(t) is an orthonormal frame for R2.

Definition II.26 (Angular Function). Let γ : I −→ R2 be a regular, parameterised curve and letV, W be an orthonormal moving frame along γ. A function α : I −→ R is called an angularfunction for γ with respect to V, W if it is continuously differentiable and, for all t ∈ I,

γ′(t)|γ′(t)| = cos

(α(t)

)V(t) + sin

(α(t)

)W(t) .

Thus, an angular function is given by the angle the tangent vector makes with respect to achosen orthonormal frame. Herein, and this will be essential later on, the frame itself is allowedto change as well. If α is an angular function, it clearly is a primitive for α′, and we have

b∫a

α′(t)dt = α(b)− α(a) .

Now if γ is a closed curve, the difference of angles at the start and at the end should be 0, but asthe angle is only unique up to addition of 2πk, k ∈ Z, we would expect α(b)− α(a) ∈ 2πZ. Thisis exactly what the following theorem says, tangent vectors of closed plane curves do full turns.

Theorem II.27 (of Turning Tangents – smooth). Let γ : [a, b] −→ R2 be a regular closedplane curve of class C2 and let α be an angular function for γ with respect to an orthonormalmoving frame V, W. Then, there is nγ ∈ Z so that

b∫a

α′(t)dt = 2πnγ . (II.13)

If γ is simple, then nγ = ±1.

The integer nγ is called the turning number of γ. The turning number of a Jordan curve is 1if and only if it is positively parameterised. Note that we have not shown that angular functionsin fact exist and this turns out to be the main part in proving Theorem II.27.

Example II.28 (The Circle). Consider the curve γ(t) =(

cos t, sin t). If t ∈ [−π, π] for instance, this willbe a simple closed curve as γ(−π) = γ(π) and γ

∣∣[−π,π)

is injective. The interior of this curve is theopen unit disc,

int(γ) =(x, y) ∈ R2 ∣∣ x2 + y2 < 1

and since γ′(t) = (− sin t, cos t) =

(cos(t + π

2 ), sin(t + π2 )), an angular function for γ with respect to

the standard frame of R2 is given by α(t) = t + π2 . Note that the tangent vector γ′(t) starts pointing

page 26


at 6 o’clock and then, turning counter clockwise, moves twelve hours to end up at 6 o’clock again. Inaccordance with this, Theorem II.27 states that

π∫−π

(t + π2 )′ dt =

π∫−π

1 dt = π − (−π) = (+1) · 2π ,

which corresponds to one full (12 hours) turn. Moreover:

Parameterising the circle in a clockwise direction, we obtain α(t) = −t + π2 and consequently a

turning number of −1. Had we taken I = [−kπ, kπ], this would have changed the limits of the integral, only, and we

would have obtained a turning number of k.

Chains and Turning Tangents

From basic geometry, we know that the sum of interior angles of a polygon with n vertices isalways given by (n− 2)π. If we define the exterior angles by π − (interior angle), this reads

∑ (exterior angles) = 2π , (II.14)

which bears at least some resemblances to Theorem II.27. We will now show in which sensethis and Theorem II.27 combine and generalise to polygons with curved edges, resulting in theTheorem of Turning Tangents, Theorem II.31.

Definition II.29 (Chains and Polygons).

i) A piecewise-Ck closed plane curve, or plane chain (of class Ck), is a continuous plane curveγ : [a, b] −→ R2 which is closed and for which there are numbers

a = t1 < t2 < · · · < tn < tn+1 = b

so that γ∣∣(ti ,ti+1)

is k-times continuously differentiable, where 1 ≤ i ≤ n.

ii) The points γ(ti) are called vertices, and the arcs γ((ti, ti+1)

)are called edges of the chain γ.

iii) If γ is a simple plane chain, we call Pγ = int(γ) a generalised polygon.

In part iii) of Definition II.29 we have already used that Jordan’s Curve Theorem holds for simpleplane chains as well, in fact, differentiability or regularity is not a necessary prerequisite as statedearlier. For Hopf’s Umlaufsatz on the other hand, this is an essential prerequisite. But first, weneed to generalise angular functions as we clearly cannot expect the tangent vector of a planechain to be a differentiable function, and then cannot expect differentiable angular functions toexist.

Definition II.30 (Exterior Angles). Let γ : I −→ R2 be a plane chain and V, W be a right-handed orthonormal frame along γ. An angular function for γ with respect to V, W is a functionα : I −→ R such that:

i) α∣∣(ti ,ti+1)

is an angular function for γ∣∣(ti ,ti+1)

with respect to V, W for all 1 ≤ i ≤ n,

page 27


ii) ϑi := limε→0

(α(ti + ε)− α(ti − ε)

)∈ [−π, π] for 2 ≤ i ≤ n and,

iii) if |ϑi| = π, then ϑi is positive if and only if γ′(ti + ε), γ′(ti − ε) is right-handed for all0 < ε < ε0 for some ε0 > 0.

The angle ϑi is called the exterior angle of the chain γ at γ(ti). The exterior angle at γ(a) = γ(b)is defined to be the (unique) representative in [−π, π] of the above limit,

limε→0

(α(t1 + ε)− α(t1 − ε)

)= ϑ1 mod 2π ,

where we use the same sign convention as above.

The exterior angles exactly capture the jumps the tangent vector of a chain makes at vertices.Integrating α′ (which is well-defined on I up to a null set) and adding the respective exteriorangles, we obtain the following piecewise-smooth version of the Theorem of Turning Tangents:

Theorem II.31 (of Turning Tangents – piecewise smooth). Let γ : [a, b] −→ R2 be aplane chain of class C2 with angular function α and exterior angles ϑi, 1 ≤ i ≤ n. Then,there is nγ ∈ Z so that

b∫a

α′(t)dt +n

∑i=1

ϑi = 2πnγ .

Proof : We only prove the case where γ is in fact simple, the general case follows similarly.Hence let γ : [a, b] −→ R2 be a simple plane chain with angular function α and exterior anglesϑi. We approximate γ by regular simple closed curves γε : [a, b] −→ R2 of class C2 so that:

γε −→ γ uniformly on [a, b], as ε −→ 0

γε(t) = γ(t) if |t− ti| > ε for all 1 ≤ i ≤ n

tr(γε) ⊂ int(γ)

That is, we smooth out the corners near the vertices and let this approximate the corners better andbetter as ε −→ 0. (Such curves can be explicitly constructed, but let us not go into the detailshere.) Now let Ui :=

t ∈ [a, b]

∣∣ |t− ti| < 2ε

and V := [a, b] \⋃i Ui. Then,

b∫a

α′ε(t)dt =n

∑i=1

∫Ui

α′ε(t)dt +∫V

α′ε(t)dt

=

(n

∑i=1

αε(ti + 2ε)− αε(ti − 2ε)

)+∫V

α′ε(t)dt

=

(n

∑i=1

α(ti + 2ε)− α(ti − 2ε)

)+∫V

α′(t)dt .

page 28


As we let ε→ 0, we have α(ti + 2ε)− α(ti − 2ε) −→ ϑi and

∫V

α′(t)dt −→b∫

a

α′(t)dt ,

where we exclude points at which α fails to be differentiable in the latter integral. Since byTheorem II.27 we have

∫ ba α′ε(t)dt = ±2π, this completes the proof.

Example II.32. If γ is piecewise linear, i.e., it parameterises the boundary of a polygon with straightedges, and positively oriented, then the angular function α is piecewise constant and from The-orem II.31 we reobtain (II.14):

b∫a

α′(t)dt +n

∑i=1

ϑi =n

∑i=1

ϑi = 2π

page 29

Chapter III

Submanifolds

In this chapter we introduce the important notion of a submanifold of Euclidean space Rn. Eventhough we will mainly consider regular surfaces in R3 in the following chapters, this extra bitof generality at the start does not make a big difference. As we will make extensive use ofmultivariate calculus, please consult Section I.2 if you are not familiar with this.

We will of course start with the basic definitions of submanifolds and then define the tangentspace of a submanifold. This space will allow us to make sense of directions in a submanifoldand the way it changes while varying the point of contact will give rise to various concepts ofcurvature. To conclude, we consider functions between submanifolds and their differentials.

Possibly the earliest hint at manifold theory was the study of non-euclidean geometries in the18th Century, the study of spaces in which Euclid’s Parallel Postulate fails. These were dividedinto two classes: hyperbolic and ellipitic geometries. In today’s terms, this corresponds to con-stant negative (hyperbolic), positive (elliptic) respectively zero (euclidean) curvature. The termmanifold comes from the German word “Mannigfaltigkeit”, used by Riemann18 in his studies togeneralise the idea of surfaces to higher dimension. Important work on generalisations, found-ations and further aspects was carried out by Poincaré19, Weyl20 and Whitney21, amongst manyothers. The resulting areas of research today are an integral part of mathematics, providing linksbetween analysis, geometry, algebra and topology.

III.1 Submanifolds of Euclidean Space

Figuratively speaking, a submanifold M of RN is a subset M ⊂ RN which locally looks like apiece of Rn, where n ≤ N. But how can we make this “definition” precise? First of all, we saythat a property P holds locally in M ⊂ RN if:

For each p ∈ M there is an open neighbourhood U of p in M such that P holds on U.

Next, we need to recall that, given two sets V ⊂ M ⊂ RN , we say that V is open in M, if there isa set V ⊂ RN so that V = V ∩M and V is open. Note that V may be open in M and closed inRN at the same time: Take M to be the closed unit ball in R3 and V = M. Then, if we let V bethe open ball centred at the origin and of radius 2, we see that V is open and V = V ∩M. Thisin fact shows that any set is open in itself!

18 Georg Friedrich Bernhard Riemann, ∗ 1826 in Breselenz, Germany, † 1866 in Selasca (Ghiffa), Italy19 Jules Henri Poincaré, ∗ 1854 in Nancy, † 1912 in Paris20 Hermann Klaus Hugo Weyl, ∗ 1885 in Elmshorn, Germany, † 1955 Zurich, Switzerland21 Hassler Whitney, ∗ 1907 in New York, † 1989 in Princeton

page 30

2016–17 · Differential Geometry Submanifolds of Euclidean Space · III.1

Definition III.1 (Submanifolds of RN ). A subset M ⊂ RN is an n-dimensional differentiablesubmanifold of RN , if it is locally diffeomorphic to Rn. That is, if for all p ∈ M, there is an openneighbourhood V ⊂ M of p, an open subset U ⊂ Rn and a map ϕ : U −→ RN so that

i) ϕ : U −→ RN is continuously differentiable,

ii) ϕ : U −→ V is a homeomorphism,

iii) Dϕ∣∣u : Rn −→ RN is injective for all u ∈ U.

The map ϕ is called a local chart, the neighbourhood V its coordinate patch. The componentfunctions of ϕ−1 =

(u1, . . . , un

)are called local coordinates.

If

Vi

is a covering of M by coordinate patches with associated charts ϕi : Ui −→ Vi, then thecollection of triples

(ϕi, Ui, Vi)

is called an atlas of M.

We will usually express elements of Rn as q = u = (u1, . . . , un), for n = 2 commonly as (u, v). InRN , we will often write ϕ(u) = p ∈ M or ϕ(u) = x = (x1, . . . , xN) ∈ M. In case N = 3, we willuse the common notation (x, y, z) ∈ M. Please be aware that we do not notationally distinguishbetween the components uj of the vector u ∈ Rn and the component functions of ϕ−1, i.e., localcoordinates uj : M ⊃ V −→ Rn.

Example III.2.i) For n ≤ N, Rn can be considered an n-dimensional submanifold of RN : The map

ι : Rn −→ RN , (u1, . . . , un) 7−→ (u1, . . . , un, 0, . . . , 0)

is a globally defined chart for Rn ⊂ RN . Similarly, any open subset of Rn can be considered ann-dimensional submanifold of RN .

ii) If I ⊂ R is open and γ : I −→ RN is a regular and simple parameterised curve, then its trace is a1-dimensional submanifold of RN with γ being a chart for tr(γ). Both regularity and injectivityare central as shows the next example.

iii) A figure eight in R2 is not a submanifold because there cannot be charts for neighbourhoods ofthe crossing point. Intuitively, any such neighbourhood consists of two curves which intersect ina single point and this cannot be homeomorphic to a subset of R. (This is not a rigorous proof!)

iv) The unit circle in R2, S1 =(x, y) ∈ R2

∣∣ x2 + y2 = 1

, is a 1-dimensional submanifold of R2:We will show that the maps

ϕ1 : (0, 2π) −→ R2 , ϕ1(t) = (cos t, sin t) ,

ϕ2 : (0, 2π) −→ R2 , ϕ2(t) = (− cos t,− sin t) .

are charts and form an atlas for S1. First of all, note that V1 = ϕ1((0, 2π)

)= S1 \ (1, 0) and

V2 = ϕ2((0, 2π)

)= S1 \ (−1, 0) indeed form an open cover of S1. Both maps are clearly

continuously differentiable and

Dϕ1∣∣t =

(− sin tcos t

), Dϕ2

∣∣t =

(sin t− cos t

)are injective, e.g.

Dϕ1∣∣t(a) =

(− sin tcos t

)· a =

(−a sin ta cos t

)= 0 ⇐⇒ a = 0 .

page 31


Moreover, ϕ1 : (0, 2π) −→ V1 is a continuous bijection and open, i.e., it maps open sets to opensets. To see this, it suffices to show that for any open interval I ⊂ (0, 2π), there is an open setO ⊂ R2 such that ϕ1(I) = O ∩ S1, a precise construction of such a set O is left as an exercise.Now, since ϕ1 is bijective, the inverse ϕ−1

1 exists and since ϕ1 is open, this inverse is continuous.Hence ϕ1 is a homeomorphism. Similarly, ϕ2 : (0, 2π) −→ V2 is a homeomorphism. All in all,we have shown that

(ϕ1, (0, 2π), V1), (ϕ2, (0, 2π), V2)

is an atlas for S1.

v) The 2-sphere S2 =(x, y, z) ∈ R3

∣∣ x2 + y2 + z2 = 1

is a 2-dimensional submanifold of R3.One can use the stereographic projections

πN : R2 −→ S2 \ N , πS : R2 −→ S2 \ S ,

where N and S denote the north and south pole, respectively, as charts. (See later exercises.)There are of course many more charts and the study of charts of S2 with different properties(conservation of area, angle or length, for instance) is at the center of cartography and was amajor impulse for the development of differential geometry.

Remark III.3 (On Definition III.1). Though we have stated that a submanifold is locally diffeomorphic to Rn, we are not yet in the

position to formulate this precisely, as we have not yet defined what it means for the functionϕ−1 : V −→ Rn to be differentiable. This will be done in Section III.3. Strengthening the condition in item i) of Definition III.1 to, say, requiring ϕ to be Ck leads to

submanifolds of class Ck, k ≥ 1. (Note that ϕ has to be differentiable for item iii) to make sense.)Choosing k = 0 and thus just requiring ϕ to be continuous and dropping item iii) altogether, weobtained a topological submanifold. But here, all submanifolds will be at least differentiable, i.e.,of class C1. Due to a theorem of Whitney any Ck-manifold, 0 < k < ∞, possesses a C∞-atlas. This is not true

for k = 0 unless n < 4: Any C0-manifold of dimension n < 4 possesses an essentially uniqueC∞-atlas. (This is was established by proving the Hauptvermutung in this case.) Item ii) makes sure that, locally, a submanifold indeed looks like a piece of Rn, the property of

“looking like” being made rigorous by the use of homeomorphisms. Item iii) ensures that the tangent space, which we will introduce later, is well-defined and of

dimension n. Please recall from linear algebra that the following are equivalent (with notationas in Definition III.1):

i) Dϕ∣∣u : Rn −→ RN is injective

ii) Jϕ∣∣u has rank n

iii) ∂ϕ∂u1

∣∣u, . . . , ∂ϕ

∂un

∣∣u are linearly independent

The vectors ∂ϕ∂uj

∣∣u will be of importance for the description of the tangent space.

Why was it important to exclude the points (1, 0) respectively (−1, 0) from the image of thecharts ϕ1, ϕ2 in Example III.2, iv)? Suppose for instance we allowed t = 0 in the definition of ϕ1.We then obtained a surjective map

ϕ1 : [0, 2π) −→ S1 ,

page 32


and ϕ1 would still be a continuously differentiable map with injective differential. But it wouldnot be a homeomorphism anymore: For small enough ε > 0, the sets [0, ε) are open neighbour-hoods of 0 in [0, 2π). However, any open neighbourhood of (1, 0) in S1 necessarily contains anelement (x, y) with y < 0 whereas ϕ1

([0, ε)

)does not. Hence ϕ−1

1 is not continuous and ϕ1 nota homeomorphism. Observe that this only shows that we have chosen the chart (or rather itsdomain) badly, it does not say anything about S1 being a submanifold or not.

Remark III.4 (On Homeomorphisms). Showing that the map ϕ is a homeomorphism onto its image is anintegral part of verifying (or falsifying) that it is a chart. Often, the difficult part is to see that ϕ−1 iscontinuous. The following two approaches can be of particular use for this:

i) Showing that ϕ is open, i.e., showing that, whenever O is open in U, then so is ϕ(O) in V. (Thisis exactly the more topological definition of ϕ−1 being continuous.)

ii) Showing that, whenever a sequence pj ⊂ V converges in V, say to a point p, then the sequenceof preimages qj = ϕ−1(pj) converges in U with limit q = ϕ−1(p). (This is the definition of ϕ−1

being sequentially continuous. In the situation of this lecture, any sequentially continuous functionis continuous.)

Any submanifold M of RN will have infinitely many atlases. For instance, if ϕ : U −→ V is achart for an open subset V of M and ψ : U′ −→ U is any diffeomorphism of open subsets of Rn,then ϕ ψ will also be a chart for V. The definition of a submanifold only asserts the existenceof at least one atlas! Just in the same way that a curve (seen as an equivalence class) has manydifferent parameterisations, a submanifold has many different charts. Assume that ϕ : U −→ Vand ϕ′ : U′ −→ V′ are two charts of a submanifold M of RN and that V = V ∩ V′ 6= ∅. Then,since ϕ and ϕ′ are homeomorphisms, we have a map

h = ϕ′−1 ϕ : ϕ−1(V) −→ ϕ′

−1(V) .

If u = (u1, . . . , un) and u′ = (u′1, . . . , u′n) denote local coordinates with respect to ϕ and ϕ′,respectively, and if p ∈ V , we have

(h u)(p) =((ϕ′−1 ϕ) ϕ−1

)(p) = ϕ′

−1(p) = u′(p) .

For this reason, h is called a change of coordinates. Since h is a composition of homeomorphisms,it is clear that h is a homeomorphism, in fact it is a diffeomorphism:

Proposition III.5. Changes of coordinates are diffeomorphisms.

We delay the proof until we have understood submanifolds a bit better, in particular until wehave seen two important ways in which we can represent submanifolds. These can also be ofgreat help when you want to show that a set M is not a submanifold, i.e., if you want to showthat there is a point p ∈ M for which there cannot possibly be a chart for a neighbourhood ofthis point. (Example III.2, item iii) sketches how to do this in a very special situation.) The fact

page 33


that the following two characterisations are indeed equivalent to Definition III.1 is a fundamentalaspect of the theory of (sub-)manifolds.

The first one characterises submanifolds as graphs. The graph of a function g : Rn1 −→ Rn2

is the set(x, y) ∈ Rn1+n2

∣∣ g(x) = y

. In the following proposition, we allow for a reorderingof the coordinate axes in Rn1+n2 , which is why permutations appear.

Proposition III.6 (Submanifolds as Graphs). A set M ⊂ RN is an n-dimensional subman-ifold of RN if and only if M is given locally as a graph of a continuously differentiable function ofn variables, more precisely, if and only if:

For each p ∈ M there is an open neighbourhood V of p in M, an open set U ⊂ Rn, apermutation π of 1, . . . , N and a continuously differentiable map g : U −→ RN−n such that

V =(xπ(1), . . . , xπ(N)) ∈ RN ∣∣ (x1, . . . , xn) ∈ U, (xn+1, . . . , xN) = g(x1, . . . , xn)

.

The second way to look at submanifolds characterises them as level sets. A level set of afunction f : Rn1 −→ Rn2 is the preimage of single point: f−1(c) =

x ∈ Rn1

∣∣ f (x) = c

wherec ∈ f (Rn1) ⊂ Rn2 .

Proposition III.7 (Submanifolds as Level Sets). A set M ⊂ RN is an n-dimensional sub-manifold of RN if and only if M is given locally as the set of solutions of a system of N − nindependent equations, more precisely, if and only if:

For each p in M there is an open neighbourhood V of p in M, an open neighbourhood W of Vin RN and a continuously differentiable map f : W −→ RN−n, f = ( f1, . . . , fN−n) such that

i) V = f−1(0) = x ∈W∣∣ f (x) = 0

and

ii) D f∣∣

p : RN −→ RN−n is surjective for all p ∈ V.

Again, the last condition in Proposition III.7 is equivalent to the gradients of the componentfunctions of f , ∇ f1(p), . . . ,∇ fN−n(p), being linearly independent at every point p ∈ V.

Example III.8 (Coordinates on the Sphere). We consider the 2-sphere again. With regard to Proposi-tion III.7, we have S2 = f−1(0) for f (x, y, z) = x2 + y2 + z2 − 1. This function f is continuouslydifferentiable and its differential D f

∣∣(x,y,z) = (2x, 2y, 2z) vanishes if and only if (x, y, z) = 0. But since

0 6∈ S2, D f∣∣

p is surjective as a linear map R3 −→ R for all p ∈ S2.In terms of Proposition III.6, the upper half-sphere is given by

Vu = (

x, y,√

1− x2 − y2) ∣∣ |(x, y)| < 1

, (III.1)

and therefore is the graph of the function (x, y) 7→√

1− x2 − y2. (We do not need a permutationhere.) The right half-sphere is

Vr = (√

1− y2 − z2, y, z) ∣∣ |(y, z)| < 1

,

page 34


and therefore the graph of the same function, using the permutation 1, 2, 3 7→ 2, 3, 1. We can coverthe sphere by six patches in this way, each of which is the graph of the same function but using adifferent permutation.

Of course, we can also use “angular coordinates” to represent the sphere. The map

ϕ : (0, π)2 −→ R3 , ϕ(α, β) =(

cos α sin β, cos β, sin α sin β)

(III.2)

gives a local chart for the upper half-sphere as well. Now compare (III.1) and (III.2): In the first, localcoordinates (cartesian coordinates, so to say) are x(p) and y(p) while in the second, the angular localcoordinates are α(p) and β(p). As we clearly have x(p) = cos α(p) sin β(p) and y(p) = cos β(p), thechange of coordinates between these charts is given by

h : (−π2 , π

2 )2 −→ D2 , h(s, t) =

(cos s sin t, cos t

), (III.3)

where D2 =(x, y) ∈ R2

∣∣ ∣∣(x, y)∣∣ < 1

denotes the open unit disc. Its differential is given by

Dh∣∣

p =

(− sin s sin t − cos s cos t

0 − sin t

),

and as det Dh∣∣

p = sin s sin2 t 6= 0 for (s, t) ∈ (0, π)2, this is a diffeomorphism (using the InverseFunction Theorem, Theorem I.10).

Proof (of Proposition III.6) : Assume that M ∈ RN is given locally as a graph. That is, forp ∈ M, there is an open neighbourhood V of p in M, an open set U ⊂ Rn and a continuouslydifferentiable map g : U −→ RN−n so that for

x = (x′, x′′) , x′ = (x1, . . . , xn) , x′′ = (xn+1, . . . , xN)

we have

V =

x ∈ RN ∣∣ x′ ∈ U , x′′ = g(x′)

.

(Without loss of generality we may assume that the permutation is the identity, this is achievedby simply relabeling the coordinate axes.) Then, we define a map

ϕ : U −→ RN , ϕ(u) =(u, g(u)

).

Since g is continuously differentiable, so is ϕ and ϕ : U −→ V is a continuous bijection.Moreover, if O ⊂ U is open, then so is O×RN−n and since

ϕ(O) =(

O×RN−n)∩V ,

we see that ϕ(O) is open as well. Thus, ϕ : U −→ V is a homeomorphism. Lastly, note that

Dϕ∣∣u =

(id

Dg∣∣u

), for all u ∈ U,

which has rank n. This is to say that Dϕ∣∣u is injective for all u ∈ U. Thus, ϕ is a local chart of M

near p and we have shown that M is an n-dimensional submanifold of RN .

Now assume that M is an n-dimensional submanifold of RN . Say, for p ∈ M, there is achart ϕ : U −→ V for an open neighbourhood V of p in M. Let q ∈ U so that ϕ(q) = p. By

page 35


assumption, Dϕ∣∣q : Rn −→ RN has rank n, without loss of generality we assume that the first

n rows of the matrix representing Dϕ∣∣q are linearly independent. Let ϕ = (ϕ′, ϕ′′), where ϕ′

consists of the first n component functions of ϕ and ϕ′′ of the last N− n. Now Dϕ′∣∣q is invertible,

and, using the Inverse Function Theorem, Theorem I.10, it is a local diffeomorphism, say as amap ϕ′ : U′ −→ U′′. Now let

g = ϕ′′ ϕ′−1 : U′′ −→ RN−n .

Then,

V′ = ϕ(U′) =

ϕ(u)∣∣ u ∈ U′

=

(ϕ′(u)ϕ′′(u)

) ∣∣∣∣∣ u ∈ U′

=

(v

ϕ′′(

ϕ′−1(v)) ) ∣∣∣∣∣ v ∈ U′′

=

(v

g(v)

) ∣∣∣∣∣ v ∈ U′′

,

which shows that, locally near p, M is the graph of g.

Proof (of Proposition III.7) : Assume that M is given locally, in an open subset V in M, as thelevel set of a function f : W −→ RN−n, where f and W are as in Proposition III.7. Then,D f∣∣

p : RN −→ RN−n is surjective for all p ∈ V and hence has rank N − n. Writing p ∈ W as

p = (u, v) = (u(p), v(p)), where u ∈ Rn, v ∈ RN−n, and suitably relabeling the coordinate axes,this is to say that the matrix

∂( f1, . . . , fN−n)

∂(v1, . . . , vN−n)

∣∣p

is invertible for all p ∈ V. Using the Implicit Function Theorem, Theorem I.12, we see thatthere are open neighbourhoods U′ of u(p) in Rn and U′′ of v(p) in RN−n and a continuouslydifferentiable function g : U′ −→ U′′ so that

f (u, v) = 0 ⇐⇒ v = g(u)

for all u ∈ U′, v ∈ U′′. But then, M is locally given as the graph of the function g.

Conversely, assume that M is locally given as a graph. Say, for an open subset V in M, thereis an open set U ⊂ Rn and a continuously differentiable function g : U −→ RN−n, so thatV =

(u, v) ∈ RN

∣∣ v = g(u)

. Then, let f (u, v) = v− g(u). Clearly, f−1(0) = V, and

D f∣∣(u,v) =

(− Dg

∣∣u , id

): RN −→ RN−n

is surjective for all (u, v) and so M is locally a level set of the function f . By Proposition III.6,the proof is completed.

Remark III.9 (What you should remember from those proofs). There are a few aspects of the proofs of Pro-positions III.6 and III.7, which you should remember. Assume we are in the situation of a surface, i.e.,n = 2 and N = 3.

page 36

2016–17 · Differential Geometry The Tangent Space · III.2

i) If M is given near p as the set of solutions to f (x, y, z) = 0, where D f∣∣

p = ∇ f (p) 6= 0, you mayfind a representation of M as a graph as follows:

Pick one of the variables x, y, z for which the partial derivative at p does not vanish, say z. Then,using the Implicit Function Theorem, the equation f (x, y, z) = 0 can be implicitly solved for zand you obtain a function g such that z = g(x, y). (The actual solving for z is of course anothermatter and might not always be straight-forward.)

ii) If M is given locally as the graph of a function g(x, y), then a local chart for M is given byϕ(u) =

(u, g(u)

).

Proof (of Proposition III.5) : We have seen in the proof of Proposition III.6 that from ϕ and ϕ′,we can construct functions g and g′ so that, in suitably small neighbourhoods W and W ′,

ϕ(W) =(x1, . . . , xN)

∣∣ (x1, . . . , xn) ∈W , (xn+1, . . . , xN) = g(x1, . . . , xn)

,

ϕ′(W ′) = (

xπ(1), . . . , xπ(N)

) ∣∣ (x1, . . . , xn) ∈W ′ ,

(xn+1, . . . , xN) = g′(x1, . . . , xn)

where π is a permutation of the set 1, . . . , N and we assume that ϕ(W) = ϕ′(W ′). But then,the change of coordinates h is given by

(x1, . . . , xn) 7−→(

x1, . . . , xn, g(x1, . . . , xn))= (y1, . . . , yN) 7−→ (yπ−1(1), . . . , yπ−1(n)) .

The first is continuously differentiable by assumption and the latter is a linear map and hencecontinuously differentiable as well. Therefore, h is continuously differentiable and since h−1 isagain a coordinate change (which we just have shown to be continuously differentiable), h isindeed a diffeomorphism.

III.2 The Tangent Space

One of the basic objects associated to a submanifold is its tangent space. One can think of it ascontaining the directions along the submanifold, which can be made precise by using curves inthe submanifold: Whenever you have p ∈ M and a parameterised curve γ : I −→ M ⊂ RN withγ(t0) = p, the tangent vector γ′(t0) will be tangent to M.

Definition III.10 (The Tangent Space). Let M ⊂ RN be a submanifold and p ∈ M. The set

Tp M :=

v ∈ RN ∣∣ ∃ ε > 0, γ : (−ε, ε) −→ M smooth : γ(0) = p, γ′(0) = v

is called the tangent space to M at p.

In fact Tp M is a vector space, spanned by ∂ϕ

∂u1

∣∣u, . . . , ∂ϕ

∂un

∣∣u

, where ϕ : U −→ V ⊂ M is a local

chart near p and u = ϕ−1(p). To see this, first note that, as a function of t, ϕ(u + tej) is a curve

in M giving rise to the tangent vector ∂ϕ∂uj

∣∣u. Moreover, for any curve γ : (−ε, ε) −→ M in M

page 37


with γ(0) = p, we have

γ′(0) =ddt

∣∣∣t=0

(ϕ ϕ−1 γ

)(t) = Dϕ

∣∣u ·

ddt

∣∣∣t=0

(ϕ−1 γ

)(t)

=n

∑j=1

∂ϕ

∂uj

∣∣u (uj γ)′(0) ,

(III.4)

whence γ′(0) is a linear combination of the tangent vectors ∂ϕ∂uj

∣∣u with coefficients given by

derivatives of local coordinates along the curve, uj γ : (−ε, ε) −→ Rn.

Remark III.11 (On the Tangent Space). The tangent space Tp M of a submanifold M at p is usually visu-alised as being attached to M at p. Nevertheless, this is just a handy visualisation, Tp M is not an affinespace in the sense that it does not consist of sums p + v, for tangent vectors v at p. In fact, it is betternot to consider sums like p + v, v ∈ Tp M at all. In the case of submanifolds of RN , this sum might bewell-defined, but elements of M and of Tp M should be considered to be objects of different types.

Since the tangent space Tp M is associated to the point p ∈ M and since, for instance, the vectors

∂ϕ

∂u1

∣∣u, . . . ,

∂ϕ

∂un

∣∣u

form a basis for Tp M (where ϕ is a local chart with ϕ(u) = p), these vectors should also be associatedto p. Or, informally, be visualised as being attached to p. Hence we will often write ∂ϕ

∂uj

∣∣p, compare

item i) of Example III.13.

Proposition III.12 (Bases of the Tangent Space). Let M be an n-dimensional submanifold ofRN and p ∈ M. Tp M is an n-dimensional linear subspace of RN . Moreover:

i) If ϕ is a local chart for M near p satisfying ϕ(u) = p, we have Tp M = ran Dϕ∣∣u and ∂ϕ

∂u1

∣∣u, . . . , ∂ϕ

∂un

∣∣u

is a basis for Tp M.

ii) If M is the graph of a function g near p = (x, g(x)) (compare Proposition III.6), then abasis for Tp M is given by

en1

∂g∂x1

(x)

, . . . ,

enn

∂g∂xn

(x)

,

where enj denotes the j-th standard coordinate vector in Rn.

iii) If M is a level set of a function f near p (compare Proposition III.7), we have

Tp M = ker D f∣∣

p .

Proof : Let ϕ be a chart for M near p such that ϕ(u) = p. We have already seen that the vectors

∂ϕ

∂u1

∣∣u, . . . ,

∂ϕ

∂un

∣∣u

page 38


span Tp M. Moreover, by assumption, Dϕ∣∣u is injective which implies that these vectors are

linearly independent and consequently form a basis for Tp M = ran Dϕ∣∣u. This also shows that

Tp M is an n-dimensional linear subspace of RN and completes the proof of item i).If M is locally the graph of a function g, a specific chart is given by ϕ(u) =

(u, g(u)

)and

the tangent vectors ∂ϕ∂uj

∣∣u are of the form given in item ii) of Proposition III.12. Note that by

assumption g : Rn −→ RN−n, which is to say that g has N − n component functions. Therefore,the vectors in item ii) really have n + N − n = N entries.

With regard to item iii), if M is locally the level set of a function f and γ a curve as in thedefinition of the tangent space, Definition III.10, we have f (γ(t)) = 0 for all t and consequently

0 = ddt

∣∣t=0( f γ)(t) = D f

∣∣γ(0)(γ

′(0)) .

Since this holds for any γ, we see that Tp M ⊂ ker D f∣∣

p. But D f∣∣

p is surjective and therefore we

have dim ker D f∣∣

p = dim RN − dim ran D f∣∣

p = N − (N − n) = n. Now since dim Tp M = n as

well, we obtain Tp M = ker D f∣∣

p.

Example III.13.i) If γ : I −→ RN is an injective, regular, parameterised curve, then γ is as well a chart for the

submanifold tr(γ). By Proposition III.12, item i), we have

Tptr(γ) = span

∂γ

∂t

∣∣∣p

=

λγ′(t0)∣∣ λ ∈ R, γ(t0) = p

.

ii) The upper unit half-sphere in R3 can be given as the graph of g(x, y) =√

1− x2 − y2, compareExample III.8. By item ii) of Proposition III.12, we have

TpS2 = span

1

0∂g∂x (x, y)

,

01

∂g∂y (x, y)

= span

10−x√

1−x2−y2

,

01−y√

1−x2−y2

,

where p = (x, y, g(x, y)). For instance,

TNS2 = span

1

00

,

010

,

where N = (0, 0, 1) again denotes the north pole.

iii) The paraboloid z = x2 + y2 is the level set of the function f (x, y, z) = x2 + y2 − z. Its differentialis given by

D f∣∣(x,y,z) = ∇ f (x, y, z) =

(2x, 2y,−1

)and, as a linear map, its kernel by

T(x,y,z)M = ker∇ f (x, y, z) =

v ∈ R3 ∣∣ ⟨(2x, 2y,−1) , v⟩= 0

=

v ∈ R3 ∣∣ v3 = xv1 + yv2= span

1

02x

,

01

2y

.

page 39

2016–17 · Differential Geometry Functions on Submanifolds · III.3

III.3 Functions on Submanifolds

So far, we have not yet defined what it means for a function defined on a submanifold, sayf : M −→ R, to be differentiable. The problem is that M is not an open subset of RN . Intuitively,we should only be concerned with derivatives in directions along the submanifold, and these arecharacterised via the tangent spaces. If we wanted to use the notion of differentiability in RN ,we would have to extend f to be defined on an open neighbourhood of M in RN and this cannotbe done in a unique and canonical manner.

But, we may use the key principle for doing analysis on (sub-)manifolds: We simply pull back theanalysis from M to open subsets of Rn by means of charts! (This also has the advantage of notusing the ambient space RN of M, which is of course essential if we wanted to consider M as anabstract topological space not embedded in RN .)

Definition III.14 (Differentiability on Submanifolds). Let Mj be nj-dimensional submanifoldsof RNj and let f : M1 −→ M2. We say that f is (continuously) differentiable at p ∈ M1 if, forcharts ϕ1 of M1 near p and ϕ2 of M2 near f (p), the composition

ϕ−12 f ϕ1

is (continuously) differentiable at ϕ−11 (p). We say that f is (continuously) differentiable on M1 if

it is (continuously) differentiable at every point p ∈ M1.

Similarly, if M1, M2 are at least Ck, we may define Ck-functions between them. The only differ-ence being, for instance, that we require the charts ϕ1, ϕ2 to be of class Ck and the respectivecomposition ϕ−1

2 f ϕ1 to be k-times continuously differentiable as well.

Let us be a bit more precise: If ϕ1 : U1 −→ V1 and ϕ2 : U2 −→ V2 are local charts for M1 nearp and for M2 near f (p), respectively, there is some open neighbourhood V1 of p in V1 such thatV2 = f (V1) ⊂ V2. Then, the composition f = ϕ−1

2 f ϕ1 acts downstairs and is related to f asshown in the following diagram:

RN1 ⊃ V1 V2 ⊂ RN2

Rn1 ⊃ U1 U2 ⊂ Rn2

w

f

u

ϕ1

w

f

u

ϕ2

Now U1, U2 are just open subset of Rn1 respectively Rn2 and we know how to decide whetherthis map is differentiable using Definition I.1 or Theorem I.4, for instance.

Please observe that for Definition III.14 to make sense, we need to show that it is independentof the choice of charts, i.e., we need to show that if ϕ1, ϕ2 are choices of charts and ϕ−1

2 f ϕ1

is continuously differentiable, then so is ϕ′2−1 f ϕ′1 for any other choice of charts ϕ′1, ϕ′2. But

this follows directly from the fact that changes of coordinates are diffeomorphisms: Assumingthat the two sets of charts have suitably overlapping images, h1 = ϕ′1

−1 ϕ1 and h2 = ϕ′2−1 ϕ2

are diffeomorphism and we have

( f )′ := ϕ′2−1 f ϕ′1 = h2 f h−1

1 .

page 40


Thus, ( f )′ is continuously differentiable if and only if f is continuously differentiable. (Wherethe prime of course refers to using the primed set of charts and not to the derivative.)

By definition, the tangent space is made up of directions of curves in M and hence it isnatural to expect the differential of a function f : M1 −→ M2 between submanifolds to actbetween tangent spaces. Think of a tangent vector v1 ∈ Tp M1 as the instantaneous velocity ofa particle whose trajectory in M1 is given by a curve γ such that γ(0) = p, γ′(0) = v1. Then,consider a second particle whose trajectory is given by the curve f γ in M2. The tangent vectorv2 = ( f γ)′(0) then encodes the instantaneous velocity of this second particle at t = 0.

Definition III.15 (The Differential). Let f : M1 −→ M2 be a differentiable map between sub-manifolds of RN1 and RN2 , respectively. For any p ∈ M and v ∈ Tp M1, let γ be a curve such thatγ(0) = p, γ′(0) = v. The map

D f∣∣

p : Tp M1 −→ Tf (p)M2 , v 7−→ D f∣∣

p(v) := ddt

∣∣t=0

(f γ

)(t)

is called the differential of f at p.

For this definition to be meaningful, we again have to show that it is well-defined. For one, weneed to show that a tangent vector v ∈ Tp M1 is indeed mapped into Tf (p)M2. But this followsdirectly from the definition of tangent spaces since, if γ is a curve representing v, then f γ is acurve representing a tangent vector in Tf (p)M2. Next, we need to show that D f

∣∣p is independent

of the curve γ which represents the tangent vector v: Let ϕ : U −→ V be a local chart near p andlet u(p) = ϕ−1(p). We have already seen that

Dϕ∣∣u : Rn −→ Tp M1

is a linear isomorphism and differentiating γ = ϕ ϕ−1 γ at t = 0 again shows that

v = γ′(0) = Dϕ∣∣u

ddt

∣∣∣t=0

(ϕ−1 γ

)and consequently

(ϕ−1 γ

)′(0) = Dϕ

∣∣−1u (v). Then,

D f∣∣

p(v) =ddt

∣∣∣t=0

(f ϕ ϕ−1 γ

)(t) = D( f ϕ)

∣∣u (

ϕ−1 γ)′(0)

=(

D( f ϕ)∣∣u Dϕ

∣∣−1p

)(v) ,

and this last term is independent of γ. Consequently, the differential D f∣∣

p of Definition III.15 isindependent of the choice of the curve representing the tangent vectors.

Proposition III.16. Let f : M1 −→ M2 be a C1-map between submanifolds and p ∈ M1.D f∣∣

p is a linear map and, if ϕ1, ϕ2 are local charts for M1 near p and for M2 near f (p), then

D f∣∣

p

(∂ϕ1

∂uj

)=

n2

∑i=1

∂ fi∂uj

(u) · ∂ϕ2

∂vi

∣∣f (p) , (III.5)

for all 1 ≤ j ≤ dim M1 and where p = ϕ1(u), n2 = dim M2 and f = ϕ−12 f ϕ1.

page 41


Proof : This is a direct calculation. We start by observing the following, which will come inhandy quite often: As ϕ−1 ϕ = id, and because the identity mapping is linear and hencesatisfies Did = id, we have Dϕ−1( ∂ϕ

∂uj

)= D(ϕ−1 ϕ)(ej) = ej, as ∂ϕ

∂uj= Dϕ(ej). Then, using the

chain rule, Theorem I.9:

D f∣∣

p

(∂ϕ1

∂uj

)= D

(ϕ2 f ϕ−1

1)∣∣

p

(∂ϕ1

∂uj

)=(

Dϕ2∣∣

f (u) D f∣∣u Dϕ−1

1

∣∣p

)(∂ϕ1

∂uj

)=(

Dϕ2∣∣

f (u) D f∣∣u

)(ej) = Dϕ2

∣∣f (u)

( ∂ f∂uj

∣∣u

)=

n

∑i=1

∂ fi∂uj

(u) · ∂ϕ2

∂vi

∣∣f (p)

This proves the local coordinate representation for D f∣∣

p. Moreover, since we have expressed D f∣∣

p

using matrix multiplication and bases of Tp M1 and Tf (p)M2, this also shows that D f∣∣

p is a linearmap.

In the past paragraphs, we have extensively used the chain rule, Theorem I.9. But observethat so far, we have it at hand for functions defined on open subsets of some Rn, only. And,without mention, we have made sure that we only use it in this way. As it is an importantcomputational tool, it is fortunate (though not surprising) that it generalises to the context ofsubmanifolds:

Proposition III.17 (Chain Rule). Let Mj, j = 1, 2, 3, be submanifolds of RNj , respectively,and f : M1 −→ M2, g : M2 −→ M3 be differentiable. Then, g f is differentiable at p ∈ M1

and its differential is given by

D(g f )∣∣

p = Dg∣∣

f (p) D f∣∣

p : Tp M1 −→ Tg( f (p))M3 .

Proof : First of all, please note that using charts near p, f (p) and g( f (p)), respectively, we seethat g f = g f , where a tilde is used as before to denote the respective composition with localcharts. Then, since D f

∣∣p maps into Tf (p)M2, the composition of D f

∣∣p and Dg

∣∣f (p) is indeed

well-defined. Lastly, let γ : (−ε, ε) −→ M1 be a curve with γ(0) = p, γ′(0) = v. Clearly,f γ : (−ε, ε) −→ M2 is a curve in M2 satisfying ( f γ)(0) = f (p) and ( f γ)′(0) = D f

∣∣p(v).

Consequently,

D(g f )∣∣

p(v) =(

g f γ)′(0) = Dg

∣∣f (p)

(( f γ)′(0)

)= Dg

∣∣f (p)

(D f∣∣

p(v))

=(

Dg∣∣

f (p) D f∣∣

p

)(v) ,

where we have used the classical chain rule, Theorem I.9, in the second step.

page 42


Example III.18 (Functions on Submanifolds).i) If γ : I −→ R3 is a regular, injective C1-parameterised curve, the unit tangent, principal unit

normal and unit binormal vectors define C1-functions,

T, N, B : tr(γ) −→ R3 .

(Here, we say they define functions on tr(γ) as they are originally defined on I instead of γ(I).)A chart for tr(γ) is clearly given by ϕ1 = γ, the corresponding tangent vector is ∂ϕ

∂s

∣∣p = γ(s0) =

T(s0) for γ(s0) = p. As a chart for R3 we may choose ψ(x, y, z) = xT(s0) + yN(s0) + zB(s0), thecorresponding tangent vectors are ∂ψ

∂x = T(s0),∂ψ∂y = N(s0) and ∂ψ

∂z = B(s0) and independent of

the point in R3.

Consider T for instance. The map “downstairs” is given by T = ψ−1 T, as T really is definedon I, already. Thus,

T(s) =(⟨

T(s) , T(s0)⟩,⟨

T(s) , N(s0)⟩,⟨

T(s) , B(s0)⟩)

,

DT∣∣s =

∂T∂s

(s0) =(⟨

T(s0) , T(s0)⟩,⟨

T(s0) , N(s0)⟩,⟨

T(s0) , B(s0)⟩)

.

This of course just returns the Frenet-Serret equation for T: DT∣∣

p = κ(s0)∂ψ∂y

∣∣p = κ(s0)N(s0).

But, T, N and B can also be considered functions tr(γ) −→ S2, as they are all of unit length.When determining DT in this situation, is there are a natural choice of chart for S2 and whatwill DT be?

ii) If M ⊂ RN is a submanifold and p0 ∈ RN \M, then the squared distance to p0,

f : M −→ R , p 7−→ |p− p0|2 ,

is C1. Its differential can be computed as follows: Let ϕ be a local chart near p ∈ M andϕ(u) = p. Since |p− p0|2 =

⟨p− p0 , p− p0

⟩, we have

Dd∣∣

p

( ∂ϕ

∂uj

)=

∂

∂uj

(⟨ϕ(u)− p0 , ϕ(u)− p0

⟩)= 2

⟨ ∂ϕ

∂uj, ϕ(u)− p0

⟩.

Then, if X = ∑j Xj∂ϕ∂uj∈ Tp M,

Dd∣∣

p(X) = 2⟨

X , p− p0⟩

.

For the sake of completeness, we close this section by mentioning that a continuously differenti-able map between submanifolds of RN1 and RN2 , say f : M1 −→ M2, is said to be

i) an immersion, if D f∣∣

p : Tp M1 −→ Tf (p)M2 is injective for all p ∈ M1,

ii) an embedding, if f : M1 −→ f (M1) is an immersion and a homeomorphism,

iii) a submersion, if D f∣∣

p : Tp M1 −→ Tf (p)M2 is surjective for all p ∈ M1

iv) and a diffeomorphism, if all of the above hold.

Observe that this definition of diffeomorphisms is equivalent to asking f and f−1 to exist andbe C1 and that for dim M1 > dim M2 there are cannot be immersions M1 −→ M2 while fordim M1 < dim M2 there are cannot be submersions M1 −→ M2.

page 43

2016–17 · Differential Geometry Surfaces in 3-Dimensional Space · III.4

III.4 Surfaces in 3-Dimensional Space

From now on, we will restrict all considerations to the special case n = 2, N = 3. By a (regular)surface, we understand a 2-dimensional submanifold of R3, which we will generally denote byΣ. Local charts for surfaces are sometimes also called parameterisations, and the tangent spacebecomes the tangent plane. As general notation, we let

R2 ⊃ U 3 q = (u, v) σ−−−−→ σ(u, v) = (x, y, z) = p ∈ V ⊂ Σ ⊂ R3

denote the standard coordinates in R2 respectively R3 and σ a local chart for a surface Σ. Wewill also regard u = u(x, y, z) and v = v(x, y, z) as local coordinates in Σ and sometimes writex = x(u, v) (and similarly for y and z) for the component functions of σ.

Tangent vectors will be denoted by X or Y, for instance, or even X∣∣

p or Y∣∣

p when we want to

emphasise the base point p. We will denote the standard unit tangent vectors to R2 by ∂1 and∂2, and use a shorthand for the tangent vectors induced by a local chart σ,

∂u = Dσ∣∣q(∂1) =

∂σ

∂u∣∣

p and ∂v = Dσ∣∣q(∂2) =

∂σ

∂v∣∣

p ,

where in the latter equations we have emphasised the base point p instead of using the pointq = σ−1(p) at which the partial derivative is evaluated.

The following have been proven before, we only formulate them again in this special situ-ation.

Dσ∣∣q : R2 −→ TpΣ is injective if and only if ∂u × ∂v 6= 0, which in turn is equivalent to at

least one of the matrices

∂(x, y)∂(u, v)

∣∣q ,

∂(x, z)∂(u, v)

∣∣q ,

∂(y, z)∂(u, v)

∣∣q

being nonsingular, compare the notation of (I.5).

If U ⊂ R2 is open and Σ =(x, y, z) ∈ R3

∣∣ (x, y) ∈ U, z = g(x, y)

, for a continuouslydifferentiable function g : U −→ R, then

10

∂g∂x (x, y)

,

01

∂g∂y (x, y)

is a basis for TpΣ, where p = (x, y, g(x, y)), compare Example III.13, item ii).

Let W ⊂ R3 be open and f : W −→ R be continuously differentiable. Then, the differentialD f∣∣

p : R3 −→ R is surjective if and only if ∇ f (p) 6= 0, or, equivalently, if at least one of

the functions ∂ f∂x , ∂ f

∂y , ∂ f∂z is nonzero at p. In that case, Σ = f−1(0) is a surface and, as in

Example III.13, item iii),

TpΣ = ker D f∣∣

p =(

span∇ f (p))⊥

=

X ∈ R3 ∣∣ ⟨X ,∇ f (p)⟩= 0

.

page 44

Chapter IV

The Fundamental Forms and Curvature

In geometry, length and angles are of particular interest. In Rn, these can be calculated usingthe scalar product: If X, Y ∈ Rn, then∣∣X∣∣ = √⟨X , X

⟩and ^(X, Y) = arccos

⟨X , Y

⟩|X| · |Y|

are the length of X and the angle between X and Y. Using this scalar product on the tangentplane (which is a linear subspace of R3), we will define angles between tangent vectors and ameasure of surface area. Then, as was the case for curves in Chapter II, we introduce a certainunit vector whose variation leads to different notions of curvature.

IV.1 The First Fundamental Form

When studying surfaces, we are mainly interested in vectors which are tangent to the surface.The tangent space is a linear subspace of R3 and we may restrict the standard (Euclidean) scalarproduct to this subspace and in this way measure the length of tangent vectors: We simplydefine

∣∣X∣∣2p =⟨

X , X⟩. (No surprises here.)

Now let σ be a parameterisation of Σ near p, let q = σ−1(p) and suppose X ∈ TpΣ is givenby a curve γ. Using local coordinates

(u(t), v(t)

)= σ−1(γ(t)), we see that (cf. (III.4))

X = γ′(0) = Dσ∣∣

p

(ddt

∣∣t=0(σ

−1 γ)(t))= u′(0)∂u + v′(0)∂v

and thus⟨X , X

⟩= u′(0)2⟨∂u , ∂u

⟩+ 2u′(0)v′(0)

⟨∂u , ∂v

⟩+ v′(0)2⟨∂v , ∂v

⟩= u′(0)2E(q) + 2u′(0)v′(0)F(q) + v′(0)2G(q) ,

(IV.1)

for functions E(q) =⟨∂u , ∂u

⟩, F(q) =

⟨∂u , ∂v

⟩and G(q) =

⟨∂v , ∂v

⟩which depend both on q and

the chosen parameterisation σ.

Definition IV.1 (The First Fundamental Form). Let Σ be a surface and p ∈ Σ. We define

Ip : TpΣ −→ R , Ip(X) :=⟨

X , X⟩

,

and call Ip the first fundamental form of Σ at p. The functions E, F and G, obtained via (IV.1) afterchoosing a local chart, are called the local components of the first fundamental form.

Motivated by (IV.1), we will also refer to the first fundamental form as

Ip = E du2 + 2F dudv + G dv2 ,

page 45

2016–17 · Differential Geometry The First Fundamental Form · IV.1

which is just a formal expression as we are not going to introduce differential forms in thiscourse. Another shorthand to write the first fundamental form is the following: If X = a∂u + b∂v,then following (IV.1) we have

Ip(X) =

(ab

)t (E FF G

) (ab

),

whence we may identify Ip with the matrix

Ip :=

(E FF G

). (IV.2)

Again, this representation depends on p and the chosen parameterisation, which is why we callit a representation in local coordinates.

Example IV.2.i) Let us consider R2 as a surface in R3 by identifying R2 with the plane z = 0 ⊂ R2. A (global)

parameterisation for R2 is for instance σ(u, v) = (u, v, 0), (u, v) ∈ R2. The tangent vectorscorresponding to this parameterisation are just the first two standard coordinate vectors in R3

and we consequently obtain I = 1 · du2 + 1 · dv2 or E = 1, F = 0, G = 1.

ii) Let C =(x, y, z) ∈ R3

∣∣ x2 + y2 = 1

be the unit cylinder. A parameterisation of a subset of Cis given by σ(u, v) = (cos u, sin u, v). We calculate

∂u =

− sin ucos u

0

, ∂v =

001

and see that the first fundamental form of C is given with respect to σ by du2 + dv2. It is strikingthat this is actually the same expression as for the standard scalar product in R2, cf. item i).

iii) Consider a part of the 2-sphere covered by the parameterisation

σ(ϑ, ϕ) =(

sin ϑ cos ϕ, sin ϑ sin ϕ, cos ϑ)

.

Since we have

∂ϑ =(

cos ϑ cos ϕ, cos ϑ sin ϕ,− sin ϑ)

, ∂ϕ =(− sin ϑ sin ϕ, sin ϑ cos ϕ, 0

),

we obtain E = |∂ϑ|2 = 1, F =⟨∂ϑ , ∂ϕ

⟩= 0 and G = |∂ϕ|2 = sin2 ϑ. Consequently, the first

fundamental form of the sphere is given at p = σ(ϑ, ϕ) by

dϑ2 + sin2 ϑ dϕ2 or Ip =

(1 00 sin2 ϑ

).

Definition IV.3 (Orthogonal Coordinates). Let σ be a local chart for a surface Σ. We say that σ

is orthogonal or that the corresponding local coordinates(u(p), v(p)

)are orthogonal coordinates

if, with this choice of chart we have F = 0. In other words, local coordinates are orthogonal if and onlyif ∂u ⊥ ∂v.

Using the first fundamental form, we can express the length of and the angle between tangentvectors X, Y ∈ TpΣ: As was noted in the introduction, the angle between vectors is given by the

page 46


scalar product. Via the polarisation identity (which is of independent importance) we can expressthe scalar product in terms of the first fundamental form,⟨

X , Y⟩= 1

2

(Ip(X) + Ip(Y)− Ip(X−Y)

),

and hence can express the angle as a function of the first fundamental form:

^(X, Y) = arccos

Ip(X) + Ip(Y)− Ip(X−Y)√Ip(X) Ip(Y)

.

For curves γ1 : I1 −→ Σ, γ2 : I2 −→ Σ, we similarly obtain

`(γ1) =∫I1

√Ip(γ′1(t)

)dt

and

^(γ1(t1), γ2(t2)

)= arccos

Ip(γ′1(t1)

)+ Ip

(γ′2(t2)

)− Ip

(γ′1(t1)− γ′2(t2)

)√Ip(γ′1(t1)

)Ip(γ′2(t2)

) ,

where γ1(t1) = γ2(t2) = p.

Remark IV.4 (Riemannian Metrics). More generally, if M is a submanifold, a Riemannian metric is a con-tinuously differentiable choice of inner product gp on Tp M for each p ∈ M, whereby we understandthat the functions

u 7−→ gij(u) := gϕ(u)

(∂ϕ

∂ui

∣∣u,

∂ϕ

∂uj

∣∣u

)(IV.3)

are C1, for some and hence any choice of local chart ϕ. The important difference is that the scalarproduct need not be given by the Euclidean scalar product of the ambient space RN and that it mayvary with p. As is the case for any inner product on a vector space, gp can be expressed as a matrix(with respect to the basis induced by ϕ):

gp(X, Y) = Xt

gij(u)n

i,j=1Y ,

where the gij(u) are defined in (IV.3) and depend on u = ϕ−1(p) and ϕ. In the case of surfaces,g11 = E, g12 = g21 = F and g22 = G, just as in (IV.2). In light of the polarisation identity, such a scalarproduct contains exactly the same information as the first fundamental form.

But there is another quantity which is of major interest in geometry, namely area. (Set asidevolume, which we do not have in 2-dimensional surfaces.) Can we express the area of subsets ofsurfaces using the first fundamental form?

Recall that the area of the parallelogram P(a, b) spanned by vectors a, b ∈ R3 is given by thecross product which in turn can be expressed as a determinant,

area P(a, b) =∣∣a× b

∣∣ =√√√√det

(⟨a , a⟩ ⟨

a , b⟩⟨

b , a⟩ ⟨

b , b⟩) .

page 47


Following the idea that the tangent plane consists of directions, i.e., infinitesimal changes, thismotivates the following definition.

Definition IV.5 (The Surface Element). Let Σ be a surface and σ : U −→ V ⊂ Σ be a localparameterisation.

i) The surface element of Σ with respect to σ is dS :=∣∣∂u × ∂v

∣∣ dudv =√

EG− F2 dudv.

ii) If O ⊂ V is open and O = σ−1(O), we define:

area O :=∫O

dS :=∫O

√EG− F2 dudv ,

where the latter is understood to be a Lebesgue integral with respect to coordinates u and v.

Example IV.6 (The surface element of the sphere). The surface element of the sphere, with respect to theparameterisation of Example IV.2, is

dS(ϑ, ϕ) =√

1 · sin2 ϑ− 02 dϑdϕ =∣∣ sin ϑ

∣∣ dϑdϕ .

Noting that the parameterisation σ maps (0, π) × (0, π) onto a half-sphere (and that the sphere isrotationally symmetric), we verify that

area S2 = 2∫

(0,π)2

dS(ϑ, ϕ) = 2π∫

0

π∫0

| sin ϑ|dϑdϕ = 2π

π∫0

| sin ϑ| dϑ = −2π cos ϑ∣∣∣π0= 4π .

Definition IV.7 (Integration over Surface Patches). Let Σ be a surface with a local parameterisa-tion σ : U −→ V ⊂ Σ and let f : V −→ R. We say that f is integrable on V if the function

( f σ)√

EG− F2 : U −→ R

is integrable on U and in this case define the integral of f over V to be∫V

f dS :=∫U

( f σ)(u, v)√

E(u, v)G(u, v)− F(u, v)2 dudv .

This definition should of course be independent of the choice of parameterisation, and fortu-nately it is:

Proposition IV.8 (On Surface Integrals). The surface integral is well-defined.More precisely, let σj : Uj −→ V, j = 1, 2, be two local parameterisations with surface

elements dS1 and dS2 and suppose f : V −→ R. Then, the surface integral∫

V f dS1 exists if andonly if the surface integral

∫V f dS2 exists, in which case they coincide:∫

V

f dS1 =∫V

f dS2

page 48


Proof : First of all, we note that the surface elements can also be expressed in terms of thedifferentials of the local parameterisations:

dS =∣∣∂u × ∂v

∣∣ dudv =√

det((Dσ)tDσ

)dudv

Let h := σ−12 σ1 be the change of coordinates for the two given parameterisations. Then, using

the chain rule, we have

Dσ1∣∣q1

= D(σ2 h)∣∣q1

= Dσ2∣∣h(q1) Dh

∣∣q1

and consequently (Dσ2) = Dσ1 Dh−1, evaluated at the appropriate points. Assuming that f isintegrable with respect to σ2, integration by substitution, Theorem I.17, yields

∫V

f dS2 =∫

U2

( f σ2)√

det((Dσ2)tDσ2

)du2dv2

=∫

U1

(( f σ2) h

)√det

((Dσ2)tDσ2

) h∣∣det Dh

∣∣ du1dv1

=∫

U1

(f σ1)

√(det Dh−1

)(det(Dσ1)tDσ1

)(det Dh−1

) ∣∣det Dh∣∣ du1dv1

=∫

U1

(f σ1)

√det(Dσ1)tDσ1 du1dv1

=∫V

f dS1 .

In particular, Proposition IV.8 shows that the area of a surface patch V ⊂ Σ is independent ofthe choice of parameterisation σ : U −→ V. Thus, the total area of a surface is independent aswell.

Remark IV.9 (On Integration over Submanifolds). We can also define the volume element for any n-dimensional submanifold of RN . This is “inspired by” the representation of the surface elementas dS =

√det(Dσ)tDσ dudv: We can take the right-hand side to define the volume element dvol. Of

course, if n > 2, the local expressions will be more complex since the tangent spaces will then be n–instead of 2-dimensional. For n = 1 on the other hand, they simplify and lead to the so-called lineelement of a curve γ : I −→ RN ,

d` := |γ′(t)| dt .

Using this, we reobtain the formula for the arc length, compare Definition II.9, and can also define theintegral of a function along a curve: If f : tr(γ) −→ R, we have∫

γ

f d` =∫I

( f γ) |γ′(t)|dt .

page 49

2016–17 · Differential Geometry Orientability and the Gauss Map · IV.2

IV.2 Orientability and the Gauss Map

Recall from Chapter II that the curvature of a parameterised curve γ is related to the secondderivative of γ. This will be similar for surfaces and hence we make the following assumption:

From now on, all surfaces will be assumed to be at least twice continuously differentiable.

A surface in R3 has a two dimensional tangent plane at every point. Since this is a subspaceof R3, there is a one dimensional normal space to every point, spanned by the unit normals.

Definition IV.10 (Gauss Maps and Orientability). Let Σ be a surface and p ∈ Σ.

i) A unit normal to Σ at p is a vector N(p) ∈ R3 such that |N(p)| = 1 and(spanN(p)

)⊥= TpΣ .

ii) A Gauss22 map for Σ is a continuously differentiable map

N : Σ −→ S2

such that, for all p ∈ Σ, N(p) is a unit normal to Σ at p.

iii) If there is a Gauss map for Σ, we call Σ orientable, and if we fix a Gauss map for Σ, we say thatΣ is oriented.

Of course there is always a unit normal at every point p ∈ Σ, to be precise, there are alwaysexactly two. The essential obstruction to choosing a Gauss map is that we want to choose thisunit normal in a continuous manner. Also, Gauss maps always exist locally: If σ : U −→ V ⊂ Σis a local parameterisation, then

N(p) =∂u × ∂v∣∣∂u × ∂v

∣∣is a Gauss map for σ(U). Gauss maps are also called unit normal (vector) fields.

Example IV.11 (Gauss Maps).i) As usual, one of our first examples is the 2-sphere. Using the representation S2 = f−1(0) for

f (x, y, z) = x2 + y2 + z2 − 1, we obtain a Gauss map for S2 by

N(p) =∇ f (x, y, z)∣∣∇ f (x, y, z)

=(x, y, z)|(x, y, z)| .

This formula in fact always yields a Gauss map, the proof of which is left as an exercise.

ii) The classical example of a surface which is not orientable is the Möbius23 strip: Take a long slipof paper and, giving it half a twist, glue the ends together to form a band. Then, if we choseunit normals continuously, they would turn by 180

while we take a full circle around the bandhence N(p) = −N(p) for all points p on the strip. This contradicts the assumption of unit lengthwhence the Möbius strip is not orientable. (This is not a rigorous proof, but the essential ideafor one.)

22 Carl Friedrich Gauß, ∗ 1777 in Braunschweig, † 1855 in Göttingen23 August Ferdinand Möbius, ∗ 1790 in Naumburg, † 1868 in Leipzig

page 50

2016–17 · Differential Geometry The Second Fundamental Form · IV.3

It seems obvious that the way the unit normal respectively the Gauss map changes as wevary p is somehow related to the idea of curvature. This leads us to consider the differential ofthe Gauss map,

DN∣∣

p : TpΣ −→ TN(p)S2 .

However, the geometry of this map allows us to view it in a more useful way: It is easy tosee, compare Example IV.11, that TN(p)S

2 actually consists of those vectors in R3 which areorthogonal to N(p), but, by definition of N(p), this is just TpΣ again.

Definition IV.12 (Weingarten Map). Let Σ be an oriented surface. The map

W∣∣

p := −DN∣∣

p : TpΣ −→ TpΣ

is called the Weingarten24 map of the oriented surface Σ.

We will later need the fact that the Weingarten map induces a quadratic form TpΣ −→ R, whichwill be the second fundamental form. To this end, we will now show that W

∣∣p is self-adjoint,

i.e., it satisfies⟨W∣∣

p(X) , Y⟩=⟨

X , W∣∣

p(Y)⟩

for all X, Y ∈ TpΣ. For then,

(X, Y) 7−→⟨

X , W∣∣

p(Y)⟩

is symmetric and bilinear and defines a quadratic form⟨

X , W∣∣

p(X)⟩.

Proposition IV.13. The Weingarten map is self-adjoint.

Proof : If we dismiss the subscript p for a moment and let σ be a local parameterisation of Σnear p, we have⟨

∂u , W(∂v)⟩=

⟨∂σ

∂u,−∂N

∂v

⟩∗=

⟨∂σ2

∂u∂v, N⟩∗∗=

⟨∂σ2

∂v∂u, N⟩∗=

⟨∂σ

∂v,−∂N

∂u

⟩=⟨∂v , W(∂u)

⟩=⟨W(∂u) , ∂v

⟩where ∗= holds because of

⟨∂u , N

⟩= 0 =

⟨∂v , N

⟩(differentiate this with respect to v resp. u)

and ∗∗= is Theorem of Schwarz-Young, Theorem I.11. Thus we have shown W∣∣

p to be self-adjoint

on the basis ∂u, ∂v and by linearity, W∣∣

p is then self-adjoint on all of TpΣ.

IV.3 The Second Fundamental Form

Now that we have shown that the Weingarten map is self-adjoint with respect to the Euclideanscalar product, we know that the map

TpΣ 3 X 7−→⟨

X , W∣∣

p(X)⟩

is symmetric and bilinear. It is of fundamental importance, hence we call it the second funda-mental form:24 Julius Weingarten, ∗ 1836 in Berlin, † 1910 in Freiburg

page 51

2016–17 · Differential Geometry The Second Fundamental Form · IV.3

Definition IV.14 (The Second Fundamental Form). Let Σ be an oriented surface. The quadraticform

IIp : TpΣ −→ R , IIp(X) :=⟨

X , W∣∣

p(X)⟩= −

⟨X , DN

∣∣p(X)

⟩is called the second fundamental form of Σ at p.

Using the same calculations which lead to (IV.1), we may write the second fundamental formin local coordinates as

IIp(X) = u′(0)2e(q) + 2u′(0)v′(0) f (q) + v′(0)2g(q) (IV.4)

where, as usual, we used a parameterisation σ near p = σ(q), we wrote X = u′(0)∂u + v′(0)∂v

and defined

e(q) :=⟨∂u , W

∣∣p(∂u)

⟩, f (q) :=

⟨∂u , W

∣∣p(∂v)

⟩, g(q) :=

⟨∂v , W

∣∣p(∂v)

⟩. (IV.5)

Consequently, we will often refer to the second fundamental form as

IIp = e(q)du2 + 2 f (q)dudv + g(q)dv2 or IIp :=

(e ff g

). (IV.6)

Note that the calculations in the proof of Proposition IV.13 give explicit formulae for the functionse, f and g:

Corollary IV.15 (Local Components of II). The local components of the second fundamentalform are given by

e(q) =⟨

∂2σ

∂u2

∣∣p , N(p)

⟩, f (q) =

⟨∂2σ

∂u∂v∣∣

p , N(p)⟩

, g(q) =⟨

∂2σ

∂v2

∣∣p , N(p)

⟩,

where q := σ−1(p).

One major advantage of dealing with self-adjoint maps is that their spectral theory turns outto be really nice. For instance, the eigenvalues are real. Without proof, we will state the followingresult from linear algebra. Afterwards, we will use it to define various notions of curvature forsurfaces.

Theorem IV.16 (On Self-Adjoint Maps). Let V be a real 2-dimensional Euclidean vectorspace and T : V −→ V be self-adjoint with respect to the scalar product on V. Then:

i) There is an orthonormal basis e1, e2 of V consisting of eigenvectors of T and the corres-ponding eigenvalues λ1, λ2 are real.

ii) If Q(v) =⟨

Tv , v⟩

denotes the quadratic form induced by T, then the eigenvalues of T aregiven by

min

Q(v)∣∣ v ∈ V, |v| = 1

and max

Q(v)

∣∣ v ∈ V, |v| = 1

.

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2016–17 · Differential Geometry Principal, Gauss and Mean Curvatures · IV.4

IV.4 Principal, Gauss and Mean Curvatures

The way the Gauss map N of an oriented surface varies should capture how the surface is curvedin space, but this variation is completely determined by the Weingarten map W

∣∣p = −DN

∣∣p and

hence by the second fundamental form IIp.Please recall that, if T : V −→ V is an endomorphism of a vector space V, its determinant

and trace do not depend on the particular representation of T as a matrix (with respect to chosenbases of V).

Definition IV.17 (Principal, Gauss and Mean Curvature). Let Σ be an oriented surface, p ∈ Σ.

i) The eigenvalues κ1 = κ1(p), κ2 = κ2(p) of W∣∣

p are called the principal curvatures of Σ at p.If κ1 6= κ2, the corresponding eigenvectors ε1, ε2 are called the principal curvature directions.

ii) The Gauss curvature of Σ at p is K := det W∣∣

p = κ1 · κ2.

iii) The mean curvature of Σ at p is H := 12 trW

∣∣p = 1

2(κ1 + κ2

).

We will in general order the principal curvatures by size, κ1 ≤ κ2. Then, by Theorem IV.16,

κ1(p) = min

IIp(X)∣∣X ∈ TpΣ, Ip(X) = 1

,

κ2(p) = max

IIp(X)∣∣X ∈ TpΣ, Ip(X) = 1

.

One important way of interpreting the second fundamental form (and also the principalcurvatures) is given by representing an oriented surface locally as a graph of a function g overits tangent space. In this setting, the second fundamental form is connected to the Hessian25 ofg. The Hessian of a twice continuously differentiable function g : R2 −→ R is defined as

Hess(g)∣∣(u,v) :=

∂2g∂u2 (u, v) ∂2g

∂u∂v (u, v)∂2g

∂v∂u (u, v) ∂2g∂v2 (u, v)

.

Proposition IV.18. Let Σ be an oriented surface and p ∈ Σ. Then, there are neighbourhoods Uof 0 in TpΣ, V of p in Σ and a twice continuously differentiable function g : U −→ R such that

V =

p + X + g(X)N(p)∣∣X ∈ U

, (IV.7)

where g satisfies g(0) = 0, Dg∣∣0 = 0 and

Xt ·Hess(g)∣∣0 · X = IIp(X) ,

for X ∈ TpΣ. Thus, locally, Σ is the graph of g over TpΣ and Wp = Hess(g)∣∣0.

Proof : The idea is to choose the coordinate axes in R3 in such a way that p = 0 and N(p) =

(0, 0, 1), i.e., such that the tangent space at p is the x-y-plane, and then invoking the ImplicitFunction Theorem, Theorem I.12, or Proposition III.6. The details are left as an exercise.25 Otto Hesse, ∗ 1811 in Königsberg (Kaliningrad), † 1874 in München

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2016–17 · Differential Geometry Principal, Gauss and Mean Curvatures · IV.4

Proposition IV.18 shows that we can always represent a surface (locally) as the graph of afunction g on its tangent plane, and do so in such a way that in fact IIp = Hess(g)

∣∣0, with

respect to the local parameterisation given by (IV.7). Consequently, we can compute the principalcurvatures using the Hessian of the function g.

Remark IV.19 (The Meaning of Curvature).i) Since the principal curvatures are the minimum and maximum of the second fundamental form

on unit vectors, κ1 = κ2 implies that IIp is constant on tangent vectors of unit length. Due to thissymmetry, such a point p is called an umbilical point.

ii) If in particular κ1 = κ2 = 0, all curvatures vanish at p and we call p a flat point.

iii) If K = κ1 κ2 > 0: Σ is situated on one side of TpΣ, p is an elliptical point.

iv) If K = κ1 κ2 < 0: Σ is situated on both sides of TpΣ, p is an hyperbolic point.

v) If K = κ1 κ2 = 0 but not both κj are zero: p is a parabolic point.

vi) If H = 12 (κ1 + κ2) = 0, Σ is called a minimal surface.

Example IV.20 (On Curvatures). Using Proposition IV.18, we easily obtain examples for different typesof (local) geometries: We choose a function g : R2 −→ R such that g(0) = 0, Dg

∣∣0 = 0 and consider its

graph near 0. In this specific situation we have K(0) = det Hess(g)∣∣0 and H(0) = 1

2 tr Hess(g)∣∣0, which

follows from Dg∣∣0 = 0 by using formulae derived in the next subsection.

i) g(x, y) = x2 − y2: The Hessian of g is calculated to be

Hess(g)∣∣0 =

(2 00 −2

)and we obtain κ1 = −2 and κ2 = 2. The surface is hyperbolic near 0 and because of κ1 + κ2 = 0,it is a minimal surface.

ii) g(x, y) = x2: Here, κ1 = 0 and κ2 = 2, the surface is parabolic near 0.

iii) g(x, y) = x2 + y2: This time κ1 = κ2 = 2, thus the surface is elliptic near the umbilical point 0.

iv) g(x, y) = x4 + y4: We obtain κ1 = κ2 = 0 and 0 is a flat point of the surface.

Remark IV.21 (Change of Orientation and Curvature). If N is a Gauss map for Σ, so is −N. Taking −Ninstead of N is equivalent to equipping Σ with the opposite orientation. This has some consequencesfor the objects we just calculated in local coordinates: If “ 7−→” denotes “changes to, when changingthe orientation”, we obtain:

I 7−→ I , N 7−→ −N , W 7−→ −WII 7−→ −II , K 7−→ K , H 7−→ −H

and similarly for the local coordinate expressions.

We close this section with Euler’s26 Formula for the second fundamental form: Let Σ be an

26 Leonhard Euler, ∗ 1707 in Basel, † 1783 in St. Petersburg

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2016–17 · Differential Geometry Calculations in Local Coordinates · IV.5

oriented surface, p ∈ Σ and suppose that κ1(p) 6= κ2(p). Then, the corresponding principalcurvature directions ε1 = ε1(p) and ε2 = ε2(p) form an orthonormal basis of TpΣ. Consequently,any unit tangent vector X ∈ TpΣ can be written as X = ε1 cos ϑ + ε2 sin ϑ for some ϑ ∈ [−π, π).Then,

IIp(X) =⟨ε1 cos ϑ + ε2 sin ϑ ,−DN(ε1 cos ϑ + ε2 sin ϑ)

⟩=⟨ε1 cos ϑ + ε2 sin ϑ , κ1ε1 cos ϑ + κ2ε2 sin ϑ

⟩= κ1 cos2 ϑ + κ2 sin2 ϑ

(IV.8)

IV.5 Calculations in Local Coordinates

In this section, we focus on writing I, W, II, K and H in terms of local coordinates. En route, weexplore the relations between these objects more closely. Suppose Σ is an oriented surface in R3

and that σ : U −→ V is a local parameterisation of Σ near p ∈ Σ. As usual, let ∂u = ∂σ∂u and

∂v = ∂σ∂v . We will use this basis to write the objects of interest in local coordinates. Let us identify

tangent vectors X ∈ TpΣ with

X = X1 · ∂u + X2 · ∂v ≡(

X1

X2

)= X ,

where X1, X2 ∈ R and “≡” means “is identified with”. We have already used this identificationto obtain matrices I, II associated to the fundamental forms (and for differentials of functions:D f = D f ) and can of course do the same for the Weingarten map W, yielding a 2× 2–matrix W.Since we will often come back to using these “tilde-expressions”, we formalise them as follows:

Definition IV.22 (Local Coordinate Expressions). Given any basis α, β of the tangent planeTpΣ of an oriented surface Σ, we use a tilde · to express the matrix associated to I, II, W with respectto this basis. For tangent vectors X ∈ TpΣ, X denotes the coefficient vector of X with respect to thebasis α, β.

Following above calculations, we see that

Xt II Y =⟨

X , W∣∣

p(Y)⟩= Xt I ˜[W∣∣p(Y)] = Xt

(I W)

Y

holds for all X, Y ∈ TpΣ. This implies II = I W, and since I is nonsingular (because the scalarproduct is nondegenerate), we obtain

W = I−1 II .

Finally, we can compute all desired quantities in local coordinates:

i) the tangent vectors ∂u := ∂σ∂u and ∂v := ∂σ

∂v

ii) the functions E, F and G (i.e., the entries of I) and the entries of the inverse matrix I−1

iii) a Gauss map N = ∂u×∂v|∂u×∂v |

iv) the functions e, f and g, as given in Corollary IV.15, i.e., the entries of II

v) the entries of W = I−1 II

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2016–17 · Differential Geometry Calculations in Local Coordinates · IV.5

vi) K = det W = det IIdet I

= eg− f 2

EG−F2

vii) H = 12 trW = 1

2Eg−2F f+Ge

EG−F2

viii) κ1 = H −√

H2 − K and κ2 = H +√

H2 − K

ix) the principal curvature directions e1, e2 satisfying Wej = κj ej

Observe that item vii) follows from the formula

tr(

AtB)= ∑

i,jAijBij ,

and item viii) follows from the fact that K = κ1κ2 and H = 12 (κ1 + κ2). Let us close this chapter

with an example which applies this section’s “road map” :

Example IV.23 (Curvature of the Cylinder). Consider the cylinder C =(x, y, z) ∈ R3

∣∣ x2 + y2 = 1

again. In Example IV.2, we already computed the tangent vectors

∂u =

− sin ucos u

0

and ∂v =

001

and the first fundamental form I = du2 + dv2 with respect to the chart σ(u, v) =

(cos u, sin u, v

)of

some subset of C. Thus, I = id. A Gauss map is given by

N(p) = − ∂u × ∂v∣∣∂u × ∂v∣∣ = −

cos usin u

0

.

Following above road map, we calculate the entries of II,

e =⟨

∂2σ

∂u2 , N⟩

= cos2 u + sin2 u = 1

f =

⟨∂2σ

∂v∂u, N⟩

= 0

g =

⟨∂2σ

∂v2 , N⟩

= 0

and the matrix W = I−1 II = II. Then, we arrive at

K =eg− f 2

EG− F2 =1 · 0− 02

1 · 1− 02 = 0

H = 12

Eg− 2F f + GeEG− F2 = 1

21 · 0− 2 · 0 · 0 + 1 · 1

1 · 1− 02 =12

and κj =12 ±

√14 − 0, i.e., κ1 = 0, κ2 = 1. Note that

W · ∂u =

(1 00 0

)(10

)= 1 and similarly W · ∂v =

(1 00 0

)(01

)= 0 ,

which shows that the principal curvature directions are ε1 = ∂v (for κ1 = 0) and ε2 = ∂u (for κ2 = 1).Since we can cover any part of the cylinder by σ (by appropriately varying its domain), we have

shown that the cylinder has Gauss curvature 0 everywhere (whence it is called flat), is paraboliceverywhere, and has mean curvature 1

2 everywhere.

page 56

Chapter V

The Theorema Egregium and the Theorem

of Bonnet

Imagine two-dimensional beings, let us call them flatlanders, which live inside a surface andhave no means of observing things outside their surface. Are these flatlanders able to determinein which kind of surface they live, only using measurements of length and angle? For instance,the 2-sphere is fundamentally different from the plane, since you cannot “flatten” the spherewithout distorting lengths. (We will prove this in this chapter.) On the other hand, if you benda piece of paper to a half-cylinder, lengths and angles will be preserved since the paper is notelastic. Flatlanders should therefore be able to distinguish whether they live on a 2-sphere or ona cylinder.

In this chapter, we will consider this intrinsic geometry more closely. In particular, we willprove two important results on the geometry of surfaces: Gauss’ Theorema Egregium and theTheorem of Bonnet27. Both are concerned with the question of how invariants determine sur-faces, the first states that the Gauss curvature is intrinsic, i.e., invariant under isometries, whilethe latter is an analogue of the Fundamental Theorem of the Local Theory of Curves: It statesthat the fundamental forms locally determine a surface up to Euclidean motions.

V.1 Isometries

When there are two surfaces on which length, angles and area are measured in the same way,their geometry should be considered equal in some sense. This is the concept of isometries:

Definition V.1 (Isometries). Let Σ1, Σ2 be surfaces with first fundamental forms I1, I2.

i) A diffeomorphism Φ : Σ1 −→ Σ2 is called an isometry, if

I1,p(X) = I2,Φ(p)(DΦ∣∣

p(X) for all p ∈ Σ1, X ∈ TpΣ1.

In this case, the surfaces Σ1 and Σ2 are called isometric.

ii) Let V1 ⊂ Σ1 be open and p ∈ V1. A map Φ : V1 −→ Σ2 is called a local isometry at p if thereis an open set V2 ⊂ Σ2 containing Φ(p) so that Φ : V1 −→ V2 is an isometry. The two surfacesΣ1 and Σ2 are called locally isometric if there are local isometries into Σ2 at every p1 ∈ Σ1

and into Σ1 at every p2 ∈ Σ2.

iii) A surface Σ is called flat, if, for all p ∈ Σ, there is a local isometry Σ −→ R2 at p.

27 Pierre Ossian Bonnet, ∗ 1819 in Montpellier, † 1892 in Paris

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2016–17 · Differential Geometry Isometries · V.1

Since isometries preserve the first fundamental form (by definition), these are exactly thosemaps which preserve length, angles and area. Clearly, if f : Σ1 −→ Σ2 is an isometry, then fis a local isometry at every p ∈ Σ1. The converse holds as well, as long as we assume f to bebijective:

Proposition V.2. Let Σ1, Σ2 be surfaces and f : Σ1 −→ Σ2 be bijective and a local isometry atevery p ∈ Σ1. Then, f is an isometry.

Proof : The idea is to use local charts and then apply the Inverse Function Theorem for thecorresponding map f . The details are left as an exercise.

Locally speaking, isometries are characterised by them preserving the local components E, Fand G of the first fundamental form. This gives us a useful criterion to check (local) isometry ofsurfaces:

Theorem V.3. Let σj : U −→ Vj ⊂ Σj, j = 1, 2 be local parameterisations of surfaces Σj andlet f := σ2 σ−1

1 : V1 −→ V2. Then, the map f is an isometry if and only if E1 = E2, F1 = F2

and G1 = G2.

Proof : Let p ∈ V1, X ∈ TpΣ1 and γ be a curve representing X. For γ(t) = σ1(u(t), v(t)), we maywrite X = u′(0)∂1,u + v′(0)∂1,v and obtain

D f∣∣

p(X) =ddt∣∣t=0

(f γ

)(t) =

ddt∣∣t=0

(f σ1

)(u(t), v(t))

=ddt∣∣t=0σ2

(u(t), v(t)

)= u′(0)∂2,u + v′(0)∂2,v ,

where ∂j,u =∂σj∂u and similarly for ∂j,v. Consequently, if the components of the first fundamental

forms, denoted by I1 and I2, coincide we obtain

I1,p(X) =

⟨u′(0)

∂σ1

∂u+ v′(0)

∂σ1

∂v, u′(0)

∂σ1

∂u+ v′(0)

∂σ1

∂v

⟩= u′(0)2E1 + 2u′(0)v′(0)F1 + v′(0)2G1

= u′(0)2E2 + 2u′(0)v′(0)F2 + v′(0)2G2

=

⟨u′(0)

∂σ2

∂u+ v′(0)

∂σ2

∂v, u′(0)

∂σ2

∂u+ v′(0)

∂σ2

∂v

⟩= I2, f (p)

(D f∣∣

p(X))

.

Since changes of coordinates are diffeomorphisms, and because f is a change of coordinates, thisis to say that f is an isometry.

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2016–17 · Differential Geometry Isometries · V.1

Now suppose that f is an isometry. Choosing X = ∂1,u, we see that v′(0) = 0 and thereforeobtain

E1 = I1,p(∂1,u)= I2, f (p)

(D f∣∣

p(∂1,u))= I2, f (p)

(∂2,u)= E2 .

Choosing X to be ∂1,v, we analogously obtain G1 = G2. Then, using any X, the above equationsyield F1 = F2.

Conversely, given open sets V1 ⊂ Σ1, V2 ⊂ Σ2, a diffeomorphism f : V1 −→ V2 and a localparameterisation σ1 : U −→ V1 of Σ1, the composition σ2 = f σ1 : U −→ V2 will always bea local parameterisation for Σ2. Using Theorem V.3, we see that f is an isometry if and only ifthe local components of the first fundamental forms of Σ1 and Σ2 (with respect to σ1 and σ2,respectively) coincide.

Example V.4 (Flat surfaces).i) In item iii) of Example IV.2, we have seen that the local components of the first fundamental

form of the unit sphere are given (in a certain parameterisation) by E = 1, F = 0 and G = sin2 ϑ.Since these are different from those of R2 (considered as a surface in R3), this shows that atleast the parameterisations we used here do not induce local isometries between open subsetsof S2 and R2. We will later see that S2 and R2 are not locally isometric. (Which is to say that wecannot “flatten parts of the sphere”.)

ii) In item ii) of Example IV.2, we have computed the local components of I for the unit cylinder Cand have shown that those are the same as for R2. Thus, the cylinder C and the plane R2 arelocally isometric.

But they are not (globally) isometric and they are not even homeomorphic: The intersection ofC with the x-y-plane cuts C into two disjoint and unbounded sets, whereas the preimage undera homeomorphism of this will be a Jordan curve in R2 which cuts R2 into one unbounded andone bounded set.

iii) Consider the (open) cones

Ka =(r, ω, z) ∈ R3 ∣∣ z = ar, r > 0

,

where (r, ω) denote polar coordinates in the plane. Using the local parameterisation

σ(r, ω) =(r cos ω, r sin ω, ar

),

we may compute the first fundamental form and see that that it is given by

Ia = (1 + a2)dr2 + r2dω2 .

Now we introduce rescaled coordinates

r = ar and ω = a−1ω ,

where a =√

1 + a2. In these rescaled coordinates – i.e., using the rescaled local parameterisationσ(r, ω) = ( r

a cos(aω), ra sin(aω), a r

a ) – the components of the first fundamental form read E = 1,F = 0 and G = r2. Thus, they are independent of a! This shows that all the open conesare, independently of their angle of aperture, locally isometric. And since the limit case a = 0corresponds to the pinched plane R2 \ 0, the cones and R2 are locally isometric.

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2016–17 · Differential Geometry Vector Fields and the Covariant Derivative · V.2

For the “flatlanders” we talked about at the beginning of this chapter, this means that, withouttravelling too far, they will not be able to tell whether they live in a plane, in a cylinder or in anopen cone since all three surfaces are flat!

To conclude, we now give a (vague) definition of the terms intrinsic geometry or intrinsicquantity.

Definition V.5 (Intrinsic Geometry – vague). A quantity defined for a surface Σ is said to beintrinsic, if it is preserved under isometries.

(Here, a “quantity” can be, for instance, a function on Σ or on Σ × Σ, a function on tangentvectors of Σ or even more complicated objects. What it means for these objects to be preserved underisometries has to be specified for each type of object whence this is not a precise definition.)

Since isometries are exactly the maps preserving the local components of the first funda-mental form, a quantity is intrinsic if and only if its expression in local coordinates is a functionof the local components E, F and G and its derivatives.

Remark V.6 (On Isometries). In general, it is a difficult task to decide whether two surface patches V1, V2

are locally isometric. If you want to show that they indeed are, it of course suffices to give an isometry.But proving that there cannot be any isometry V1 −→ V2 is indeed hard. One usually looks to findinvariants which differ for V1 and V2: These are objects (e.g. numbers, groups, vector spaces) associatedto surface patches for which it is known that they are the same in case the patches are isometric. Thus,if you can prove they differ, you have shown that there is no isometry.

Theorem V.3 seems to answer this question, but please note that it covers a special case only: The(potential) isometry is given as a change of coordinates and the problem is shifted to deciding whetherthere are local coordinates for which the local components of the first fundamental forms are equal.

V.2 Vector Fields and the Covariant Derivative

We have already seen some important examples of vector fields, the tangent vectors ∂u and ∂v,for instance. In the following, V will always denote an open subset of a surface Σ.

Definition V.7 (Vector Fields). A (tangential) vector field of class Ck on a surface patch V ofclass Ck is a k-times continuously differentiable function

X : V −→ R3

such that X(p) ∈ TpΣ for all p ∈ V. We denote the set of Ck-vector fields on V by Xk(V).

The vector fields we consider will usually be tangent, if not, this is stated explicitly. Also, ifthe regularity k of a vector field is not mentioned, it is assumed to be at least twice continuouslydifferentiable. Mostly, we will write vector fields using a local parameterisation σ : U −→ V,

X(p) = Xu(p)∂u∣∣

p + Xv(p)∂v∣∣

p . (V.1)

page 60


Thus, even though X(p) ∈ R3, we consider it to have two components only: X ≡(Xu, Xv

). By

definition of differentiability, a tangent vector field X is k-times continuously differentiable ifand only if the local components Xu, Xv are in Ck(U).

There are two ways in which we can combine a function f ∈ Cl(V) with a vector fieldX ∈ Xk(V): We can multiply X by f ,

f X ∈ Xm(V) ,(

f X)(p) := f (p)X(p) ,

where m = mink, l, and for l ≥ 1 we can differentiate f with respect to X,

X f ∈ Cn(V) ,(X f)(p) := D f

∣∣p

(X(p)

),

where n = mink, l − 1. At p, this is just the directional derivative of f in direction X(p).(Note that if f ∈ Cl(V, W) for a second surface patch W, then X f is an n-times continuouslydifferentiable function on V with values in tangent spaces of W, but we will not consider thissituation.)

Can we also differentiate a vector field Y with respect to another vector field X? Indeed, if X,Y ∈ X1(V), then, as Y : V −→ R3, we have DY

∣∣p : TpV −→ TY(p)R

3 and may define(∇R3

X Y)(p) := DY

∣∣p

(X(p)

). (V.2)

The problem is that this does not necessarily yield a tangent vector! To see this, assume thatthere is a curve γ : (−1, 1) −→ V so that X(γ(t)) = Y(γ(t)) = γ′(t) and γ(0) = p. Then(

∇R3

X Y)(p) =

ddt

∣∣∣t=0

Y(γ(t)

)= γ′′(0) (V.3)

and as we have seen in Chapter II, this need not be tangent to the curve (or even the surface).However, in oriented surfaces, we can project onto the tangent plane by simply subtracting thenormal part of ∇R3

X Y:

Definition V.8 (Covariant Derivative). Let Σ be an oriented surface and X, Y ∈ Xk(Σ). Thecovariant derivative of Y with respect to X is the vector field ∇XY ∈ Xk−1(Σ) defined by(

∇XY)(p) :=

(∇R3

X Y)(p)−

⟨(∇R3

X Y)(p) , N(p)

⟩N(p) .

(The covariant derivative is a so-called connection, a means to differentiate vector fields—or sec-tions of bundles—on arbitrary manifolds. It is the torsion-free Levi-Civitá28-connection in thissetting.) For future reference, we collect some rather algebraic properties of the covariant deriv-ative:

Lemma V.9. Let X, Y, X1, X2, Y1, Y2, Z ∈ Xk(Σ) and f ∈ Ck(Σ).

i) ∇X1+X2Y = ∇X1Y +∇X2Y and ∇ f XY = f∇XY

ii) ∇X(Y1 + Y2) = ∇XY1 +∇XY2 and ∇X( f Y) = (X f )Y + f∇XY

iii) X(⟨

Y , Z⟩) =

⟨∇XY , Z

⟩+⟨Y ,∇XZ

⟩28 Tullio Levi-Civitá, ∗ 1873 in Padova, † 1941 in Roma

page 61


Proof : The map (X, Y) 7−→ DY∣∣

p(X) is linear in its first variable, and a derivation in its second.

The latter is to say that it is additive, D(Y1 + Y2)∣∣

p(X) = DY1∣∣

p(X) + DY2∣∣

p(X), and satisfies aproduct law

D( f Y)∣∣

p(X) = D f∣∣

p(X)Y + f DY∣∣

p(X) .

This directly implies items i) and ii) for ∇R3. Then, because the scalar product is bilinear, this

carries over to the covariant derivative. For item iii), note that

X(⟨

Y , Z⟩)

= D(⟨

Y , Z⟩)(X) =

⟨DY(X) , Z

⟩+⟨Y , DZ(X)

⟩=⟨∇R3

X Y , Z⟩+⟨Y ,∇R3

X Z⟩

.

Since Y and Z are tangent vector fields, we can exchange ∇R3for ∇, without changing the

formula, since their difference is orthogonal to TpΣ:⟨∇R3

X Y , Z⟩=⟨∇XY + λN , Z

⟩=⟨∇XY , Z

⟩,

where λ =⟨∇R3

X Y , N⟩. This proves item iii).

Seen from a more structural point of view, Lemma V.9 states that the covariant derivative

∇ : Xk(Σ)×Xk(Σ) −→ Xk−1(Σ) , (X, Y) 7−→ ∇XY

is Ck(Σ)-linear in its first variable (item i)), a derivation on X(Σ) in its second variable (item ii))and compatible with the scalar product and hence the first fundamental form (item iii)).

Example V.10 (The Covariant Derivative on the Torus). Let us consider a parameterisation

σ : U = (0, 2π)2 −→ V , (u, v) 7−→((2 + cos u) cos v, (2 + cos u) sin v, sin u

),

of a part of the torus with radii 1 and 2. The canonical tangent vector fields are

∂u =

− sin u cos v− sin u sin v

cos u

, ∂v =

−(2 + cos u) sin v(2 + cos u) cos v

0

. (V.4)

Note that we have written these vector fields as functions on U, though we should really considerthem to be functions on V. (Strictly speaking, the vector fields in (V.4) are ∂u σ and ∂v σ.) However,we may calculate the covariant derivatives of pairs of such vector fields:

∇R3

∂u∂u = D(∂u)

(∂u)= D(∂u)

(Dσ(e1)

)= D

(∂u σ

)(e1) (V.5)

=

− cos u cos v sin u sin v− cos u sin v − sin u cos v− sin u 0

( 10

)=

− cos u cos v− cos u sin v− sin u

,

as ∂u = Dσ(e1). (Of course, this is exactly ∂2σ∂u2 .) Similarly

∇R3

∂u∂v = ∇R3

∂v∂u =

sin u sin v− sin u cos v

0

, ∇R3

∂v∂v =

−(2 + cos u) cos v−(2 + cos u) sin v

0

.

page 62


But in order to compute the true covariant derivatives we will also need a Gauss map. We take

N =∂u × ∂v∣∣∂u × ∂v

∣∣ = − cos u cos v

cos u sin vsin u

.

Then, ⟨∇R3

∂u∂u , N

⟩= 1 ,

⟨∇R3

∂v∂u , N

⟩= 0 ,

⟨∇R3

∂v∂v , N

⟩= (2 + cos u) cos u ,

and consequently:

∇∂u ∂u = 0 , ∇∂u ∂v = ∇∂v ∂u = − sin u2 + cos u

∂v

∇∂v ∂v = −(2 + cos u)

cos v− cos2 u cos vsin v− cos2 u sin v− sin u cos u

= (2 + cos u) sin u ∂u

The geometric interpretation is as follows: If we follow the vector field ∂u in the direction it is pointing,it will only turn in directions normal to the surface whence ∇∂u ∂u = 0. If we follow it in the directionin which ∂v is pointing, it might not be turning at all (sin u = 0 on the outer and inner equator) or itwill turn in the direction of ±∂v (top and bottom line, for instance).

We now want to write ∇ in local coordinates. As in Example V.10, we use the followingconvention from now on:

Whenever there are indices i, j, k and l whose range is not determined, they are assumed to beeither one of u or v (or the respective local coordinates).

This will simplify notation quite a bit! Now given a local parameterisation σ, we may write anytwo vector fields X, Y ∈ X(Σ) using ∂u and ∂v as in (V.1). But then, by Lemma V.9, it is easy tosee that ∇XY is a linear combination of the covariant derivatives ∇∂i

∂j. (The coefficients of thislinear combination will depend on the Xi, Yj and their first order partial derivatives, compare(V.6).) But the vector fields ∇∂i

∂j are tangent vector fields again (by definition) and hence againlinear combinations of the canonical tangent vector fields ∂u and ∂v. As the coefficients of thelatter linear combinations are quite important, they have a name:

Definition V.11 (Christoffel Symbols). Let σ : U −→ V ⊂ Σ be a local parameterisation of asurface Σ. The Christoffel29 symbols of Σ resp. ∇ (of the second kind) with respect to σ are thefunctions Γk

ij : U −→ R defined via

∇∂i∂j = ∑

kΓk

ij ∂k .

The Christoffel symbols of the first kind are defined as

Γij,k :=⟨∇∂i

∂j , ∂k⟩= ∑

lΓl

ij⟨∂l , ∂k

⟩.

29 Elwin Bruno Christoffel, ∗ 1829 in Monschau (Eifel), † 1900 in Strasbourg

page 63


Using the Christoffel symbols (of the second kind) we can express covariant derivatives inlocal coordinates. For instance, we write ∇∂u ∂v = Γu

uv∂u + Γvuv∂v. We can do the same for general

∇XY, but this will not be very instructive: If Y = Yu∂u + Yv∂v, then using Lemma V.9 we get

∇∂u Y = ∇∂u

(Yu∂u

)+∇∂u

(Yv∂v

)= ∂

∂u Yu∂u + Yu∇∂u ∂u +∂

∂u Yv∂v + Yv∇∂u ∂v

=(

∂∂u Yu + YuΓu

uu + YvΓuuv

)∂u +

(∂

∂v Yv + YvΓvuv + YuΓv

uu

)∂v ,

(V.6)

and similarly for ∇XY if we write X = Xu∂u + Xv∂v.

Also observe that, for instance, ∇R3

∂u∂v is just the partial derivative of ∂v with respect to u,

i.e., ∇R3

∂u∂v = ∂2σ

∂u∂v as we have seen in (V.5). Then, since ∂2σ∂i∂j =

∂2σ∂j∂i , this shows that Γk

ij = Γkji. In

particular, we have

∇∂i∂j = ∇∂j

∂i , (V.7)

though in general ∇XY 6= ∇YX. (But when, for instance, X = ∂ f∂u and Y = ∂ f

∂v for a functionf : U −→ Σ, we will again have ∇XY = ∇YX.)

Example V.12 (Christoffel Symbols on the Torus). Continuing from Example V.10, we can read off theChristoffel symbols for the covariant derivative on the torus, with respect to the parameterisationσ: If i 6= j, we have

∇∂u ∂u = 0 =⇒ Γiuu = 0

∇∂i∂j = −

sin u2 + cos u

∂v =⇒ Γuij = 0 , Γv

ij = −sin u

2 + cos u∇∂v ∂v = (2 + cos u) sin u∂u =⇒ Γu

vv = (2 + cos u) sin u , Γvvv = 0 .

Recall that we used a Gauss map N to define the covariant derivative: We needed to subtractthe normal part off of ∇R3

X Y. But surprisingly, the covariant derivative does not depend on thechoice of Gauss map at all, and even more only depends on the first fundamental form. (So tosay, it neither matters how we represent the normal part as long as it is not there, nor depends onlocal parameterisations in any way.) To prove this, we first show that the Christoffel symbols ofthe first kind depend on the components of the first fundamental form only:

Proposition V.13. Let i, j and k denote either u or v and Γij,k be the Christoffel symbols of thefirst kind. Denoting by Iij the entries of the matrix associated to the first fundamental form withrespect to a local parameterisation, we have

Γij,k =12

( ∂

∂iIjk −

∂

∂kIij +

∂

∂jIki

).

page 64


Proof : Using item iii) of Lemma V.9, we see that

∂

∂iIjk = ∂i

(⟨∂j , ∂k

⟩)=⟨∇∂i

∂j , ∂k⟩+⟨∇∂i

∂k , ∂j⟩

= Γij,k + Γik,j ,

and consequently Γij,k = ∂∂i Ijk − Γik,j. If we applied this equation to the term −Γik,j again,

we would end up with Γij,k = Γij,k which does not lead anywhere. But, using the symmetryΓij,k = Γji,k, we may as well write Γij,k =

∂∂i Ijk − Γki,j and then apply the identity again:

Γij,k =∂

∂iIjk − Γki,j

=∂

∂iIjk −

( ∂

∂kIij − Γkj,i

)=

∂

∂iIjk −

∂

∂kIij +

( ∂

∂jIki − Γji,k

)=

∂

∂iIjk −

∂

∂kIij +

∂

∂jIki − Γij,k

Bringing Γij,k on one side and dividing by 2 proves the claim.

By now showing that the Γkij can be expressed in terms of the Γij,k and the Iij, we complete the

task of proving that the Christoffel symbols and the covariant derivative are intrinsic.

Lemma V.14. If k 6= k, then for all i, j, k we have:

Γkij =

1det I

(Γij,k Ikk − Γij,k Ikk

)

Proof : First of all, note that k is either u or v, so k is uniquely determined by demanding k 6= k.Moreover, since the scalar product is positive definite, we know that Ikk, Ikk and det I are allstrictly positive. Now by definition, we have Γij,k =

⟨∇∂i

∂j , ∂k⟩= Γk

ijIkk + Γki jIkk and then

ΓkijIkk = Γij,k − Γk

i jIkk

= Γij,k − Γki jIkk

(Ikk I−1

kk

)= Γij,k −

(Γij,k − Γk

ijIkk

)(Ikk I−1

kk

)= Γij,k − Γij,k Ikk I−1

kk + ΓkijI

2kk I−1

kk .

Rearranging for Γkij to be on one side of the equation and multiplication by Ikk yields the claim:

Γkij det I = Γij,k Ikk − Γij,k Ikk .

page 65

2016–17 · Differential Geometry The Theorems of Gauss and of Bonnet · V.3

Corollary V.15. The covariant derivative is invariant under isometries and hence an intrinsicquantity. More precisely, if f : Σ −→ Σ is an isometry and if Γk

ij and Γkij are the Christoffel

symbols of Σ and Σ with respect to parameterisations σ and σ = f σ, then

Γkij(p) = Γk

ij(

f (p))

.

Remark V.16.i) Let us clarify what it means that the covariant derivative is preserved under isometries. Given

a diffeomorphism f : Σ −→ Σ, we transform a vector field X ∈ X(Σ) to one defined on Σ bymeans of the differential of f . This is called the push-forward of X along f and denoted by f∗X:

For q = f (p), let(

f∗X)(q) := D f

∣∣p

(X(p)

)∈ TqΣ .

This defines a vector field f∗X ∈ X(Σ). Then, Corollary V.15 states that for all X, Y ∈ X(Σ), andif f denotes the chosen isometry,

f∗(∇XY

)= ∇ f∗X f∗Y .

On the other hand, for a function F : Σ −→ R we have the pull-back

f ∗ F(p) := (F f )(p) for p ∈ Σ,

which defines a function on Σ. Then, we say that for instance Gauss curvature is preservedunder the isometry f : Σ −→ Σ, if f ∗K(p) := K

(f (p)

)= K(p) for all p ∈ Σ. (One or both of the

operations of push-forward and pull-back can be defined for all “objects” of interest and lead tothe various definitions of “being preserved under isometries”.)

ii) The definition of the Christoffel symbols of the first kind from those of the second kind followsthe general principle of lowering indices by means of the scalar product (or a Riemannian metric):We have

Γij,k = ∑l

Γlij⟨∂l , ∂k

⟩= Γl

ij⟨∂l , ∂k

⟩,

where in the last step we made use of the Einstein30 summation convention (and take sumsover all indices which appear both as a super– and as a subscript).

V.3 The Theorems of Gauss and of Bonnet

We are now prepared to prove the main result of this chapter, Gauss’ Theorema Egregium(which is Latin for Remarkable Theorem). We will also explain how to obtain the Theorem ofBonnet, though we will not give the details. As mentioned in the introduction to this chapter,we want to show that the Gauss curvature K = det II

det Idoes depend on the first fundamental form

only. But this would imply that there is a relation between I and II which holds on any surfaceand which we have not seen yet.

30 Albert Einstein, ∗ 1879 in Ulm, † 1955 in Princeton

page 66


So we ask: What are sufficient conditions on 2× 2-matrices I and II, so that they are localcomponents of the fundamental forms of a surface? Clearly, necessary conditions are that I andII have to be symmetric and that I be positive definite (i.e., E, G and EG − F2 are all strictlypositive). As will become clear in an instant, from now on

all surfaces are assumed to be at least thrice continuously differentiable!

The starting point will be to decompose derivatives of vector fields into their tangential andnormal parts:

Lemma V.17. Let σ be a local parameterisation of an oriented surface and N be its Gauss map. Then,if X ∈ X(Σ),

i) ∂2σ∂i∂j = ∇∂i

∂j + IIijN and

ii) the decomposition of ∇R3

∂iX into tangential and normal part is given by

∇R3

∂iX = ∇∂i

X +⟨

X , W(∂i)⟩

N .

Proof : In fact, since ∂2σ∂i∂j = ∇

R3

∂i∂j, the first formula is a special case of the second, for X = ∂j. To

prove the second formula, the tangential part is given (by definition) by the covariant derivativewhereas the normal part is given by

⟨∇R3

∂iX , N

⟩N. But since

⟨X , N

⟩= 0, we obtain

0 = ∂i⟨

X , N⟩=⟨∇R3

∂iX , N

⟩+⟨

X , DN(∂i)⟩=⟨∇R3

∂iX , N

⟩−⟨

X , W(∂i)⟩

,

which completes the proof.

Now, using Lemma V.17, we compute the tangential part of the third order derivatives of σ:

∂

∂i∂2σ

∂j∂l=

∂

∂i∇∂j

∂l +∂

∂i(IIjl N

)=[∇∂i∇∂j

∂l +⟨∇∂j

∂l , W(∂i)⟩

N]+[( ∂

∂iIIjl)

N + IIjl∂

∂iN]

=[∇∂i∇∂j

∂l − IIjl ·W(∂i)]+[

. . .]

N

Then, doing the same with i and j interchanged and taking the difference, we obtain

0 =∂

∂i∂2σ

∂j∂l− ∂

∂j∂2σ

∂i∂l

=[∇∂i∇∂j

∂l − IIjl ·W(∂i) +∇∂j∇∂i

∂l − IIil ·W(∂j)]+[

. . .]

N , (V.8)

and therefore

∇∂i∇∂j

∂l −∇∂j∇∂i

∂l = IIjlW(∂i)− IIilW(∂j) . (V.9)

The latter is a tangential vector field and can hence again be written as a linear combination of∂u and ∂v. The coefficients are the components of the Riemann curvature tensor:

page 67


Definition V.18 (Riemann Curvature Tensor). Let i, j, k, l denote either one of u or v. Thenumbers Rk

lij, defined by

∇∂i∇∂j

∂l −∇∂j∇∂i

∂l = ∑k

Rklij∂k ,

are called components (of the second kind) of the Riemann curvature tensor with respect to σ.The components of the first kind are defined as

Rklij :=⟨∇∂i∇∂j

∂l −∇∂j∇∂i

∂l , ∂k⟩

.

In terms of the Riemann curvature tensor, (V.9) shows that, for all i, j, k, l,

Rklij = IIjl⟨∂k , W(∂i)

⟩− IIil

⟨∂k , W(∂j)

⟩= IIjl IIik − IIil IIjk . (V.10)

The equations (V.10) are called the Gauss equations.

Theorem V.19 (Theorema Egregium). The Gauss curvature is invariant under isometries,i.e., it is an intrinsic quantity.

Proof : The Gauss curvature is given by K = det IIdet I , which, by (V.10) for (klij) = (vuvu), is given

by Rvuvudet I . Thus, if we were to show that the Riemann curvature tensor is intrinsic, we would also

have shown that the Gauss curvature is intrinsic. But the components of the Riemann curvaturetensor are given by the Christoffel symbols and their derivatives. As these are intrinsic byCorollary V.15, the Gauss curvature is an intrinsic quantity.

One of the many consequences of Theorem V.19 is that we now have a practical necessarycondition for isometry of surfaces: If two surfaces are isometric, say there is an isometry f :Σ −→ Σ, then their Gauss curvatures have to be related by K(p) = K(q), where q = f (p).Conversely, if the Gauss curvatures of two surfaces cannot be related in this way (for instancebecause the ranges differ, K(Σ) 6= K(Σ)) the surfaces cannot possibly be isometric!

Example V.20 (‘You cannot flatten a sphere!’). Let us return to the question of whether there is a map of(a subset of) the surface of Earth (assuming it is a sphere) which does neither distort angles or lengths,compare Example V.4, item i). Using the parameterisation

σ(ϑ, ϕ) = r(

sin ϑ cos ϕ, sin ϑ sin ϕ, cos ϑ)

,

where r > 0 is the radius of the sphere under consideration, we obtain the following formulae for thefundamental forms:

I(ϑ, ϕ) =

(r2 00 r2 sin2 ϑ

), II(ϑ, ϕ) =

(r 00 r sin2 ϑ

)Thus, the Gauss curvature of a sphere of radius r is given by

Kr =det IIdet I

=r2 sin2 ϑ

r4 sin2 ϑ=

1r2 .

page 68


Since the Gauss curvature of any open subset of R2 is 0, and since this quantity is preserved underisometries, there is no isometry from an open subset of R2 to an open subset of a sphere. Consequently,there is no way to draw a (plane) map of a part of the surface of Earth without perturbing angles orlengths.

So far, we have only used partial information from equation (V.8): The Gauss equations followfrom the fact that the tangential part of (V.8) vanishes. Of course, the normal part has to vanishas well, the resulting equations are called the Codazzi31-Mainardi32 equations:

∑k

(Γk

jl IIik − Γkil IIjk

)+

∂

∂iIIjl −

∂

∂jIIil = 0 (V.11)

Together with the Gauss equations, these yield a necessary and sufficient condition for the ex-istence of a surface with prescribed fundamental forms. In fact, the fundamental forms locallydetermine any surface up to a Euclidean motion. (A proof of the following result can be foundin [dC1], for instance.)

Given continuously differentiable matrix-valued functions I, II on an open subset O of R2, wecan use Proposition V.13 and Lemma V.14 to formally define Christoffel symbols, and formallydefine the components of the Riemannian curvature tensor by Definition V.18, expressing thecovariant derivatives in terms of the formal Christoffel symbols. Thus, having such functionsI and II, we may talk about whether they solve the Gauss and Codazzi-Mainardi equations(V.10) – (V.11) or not.

Theorem V.21 (of Bonnet). Let O ⊂ R2 be open and suppose I, II : O −→ R2×2 are continu-ously differentiable, matrix-valued functions which satisfy the following:

i) for any q ∈ O, I(q) and II(q) are symmetric

ii) for any q ∈ O, I(q) is positive definite

iii) I and II satisfy the Gauss equations (V.10) and the Codazzi-Mainardi equations (V.11)

Then, for each q ∈ O, there is an open neighbourhood U of q in O and a local chart

σ : U −→ σ(U) ⊂ R3 ,

such that the surface Σ = σ(U) has first and second fundamental forms given by I and II,respectively.

Moreover, if U is connected and if σ′ is another such chart, then there is a Euclidean motionR : R3 −→ R3 such that σ′ = R σ.

31 Delfino Codazzi, ∗ 1824 in Lodi, † 1873 in Pavia32 Gaspare Mainardi, ∗ 1800 in Abbiategrasso (Milano), † 1879 in Lecco

page 69

Chapter VI

Curves on Surfaces

In this chapter, we will focus on the action of the covariant derivative along curves. In particular,we look at the covariant derivative of vector fields with respect to the tangent vector of a curve.This leads to the notion of vector fields which are parallel along a curve and to the important classof curves called geodesics. For an arbitrary surface, geodesics play the role which is being playedby straight lines in R2.

VI.1 Parallel Vector Fields

Definition VI.1. Let Σ be a surface and γ : I −→ Σ be a regular parameterised curve. A vectorfield along γ, of class Ck, is a k-times continuously differentiable map

X : I −→ R3 so that X(t) ∈ Tγ(t)Σ, for all t ∈ I.

In this case, we write X ∈ Xk(γ).

Note that we view X as a function of t, not as a function of p = γ(t) as we usually do withvector fields, and that X need not be tangent to γ but only to Σ. If Y ∈ Xk(Σ), and σ is a localparameterisation near γ(t), then clearly Y σ is a vector field along γ near γ(t).

Differentiation of X ∈ Xk(γ), k ≥ 1, with respect to t givesdXdt

= ∇R3

γ′ X ,

and, if Σ is orientable, we may project this onto the tangent space of Σ to obtain yet anothervector field along γ:

∇dt

X := ∇γ′X = ∇R3

γ′ X−⟨∇R3

γ′ X , N⟩

N ∈ Xk−1(γ) . (VI.1)

Please observe that if there are vector fields X, Y ∈ X(Σ) and a local parameterisation σ so thatX = X

∣∣tr(γ) σ and γ′ = Y

∣∣tr(γ) σ, then ∇dt X(t) =

(∇YX

)(γ(t)

).

Definition VI.2 (Parallel Vector Fields). Let γ be a curve in a surface Σ.

i) The map ∇dt : Xk(γ) −→ Xk−1(γ) is called the covariant derivative along γ.

ii) If ∇dt X(t) = 0 for all t ∈ I, X is said to be parallel along γ.

Thus, we call X parallel along γ, if the variation of X, as we move along γ, cannot be seenfrom inside of Σ (or, more precisely, is normal to Σ). We have already seen parallel vector fields,for instance in Example V.10.

page 70

2016–17 · Differential Geometry Parallel Vector Fields · VI.1

Example VI.3 (Parallel Vector Fields).i) In R2 (as a surface in R3), we have ∇dt X(t) = X′(t) for any curve γ : I −→ R2 and any X ∈ X(γ).

Consequently, parallel vector fields in R2 are exactly the constant vector fields, independentlyof the curve.

ii) Consider the parameterisation σ of a subset of the torus from Example V.10 and let

γu(t) = σ(t, π) , γv(t) = σ(π, t) .

γu and γv are simple, parameterised curves on the torus, γu runs along the intersection of thetorus with the set y = 0, x < 0, γv runs along the inner equator. Clearly, γ′u = ∂u and γ′v = ∂v,and, following our calculations of the Christoffel symbols in this situation, ∂u is parallel alongboth γu and γv, whereas ∂v is parallel along γv but not along γu. Please observe two subtleties:

Even though ∂u is parallel along γu, the vectors ∂u∣∣

p, ∂u∣∣q for different points p, q ∈ tr(γu)

are in general not parallel in R3. If we had not set u = π in the definition of γv but to some other constant u0 ∈ (0, 2π), ∂u

and ∂v would not be parallel along γv anymore.

The algebraic properties of the covariant derivative on the surface carry over to the covariantderivative along curves. For instance, we have a product law for differentiable functions,

∇dt(

f X)=( d

dtf)

X + f∇dt

X (VI.2)

and compatibility with the scalar product,

ddt(⟨

X , Y⟩)

=⟨∇

dt X , Y⟩+⟨

X , ∇dt Y⟩

, (VI.3)

where X, Y ∈ X(γ). Since the property of being parallel along a curve is essentially a system oflinear, first order ordinary differential equations, there is always a unique parallel vector field aslong as we fix initial conditions.

Proposition VI.4. Let γ : I −→ Σ be a parameterised curve in a surface Σ and let t0 ∈ I,X0 ∈ Tγ(t0)

Σ. Then, there is a unique vector field X along γ, which is parallel along γ andsatisfies X(t0) = X0.

Proof : We assume that tr(γ) is contained in a single surface patch σ : U −→ V ⊂ Σ. If not,we may cover tr(γ) by multiple patches and extend X across the overlaps, as we will obtainuniqueness of X on these overlaps.

Using local coordinates γ(t) = σ(u(t), u(t)

), we may write X = Xu(t)∂u + Xv(t)∂v and

γ′(t) = γ′u(t)∂u + γ′v(t)∂v, where γ′u(t) = u′(t) and γ′v(t) = v′(t). Then,

∇γ′∂j = ∑i

γ′i(t)∇∂i∂j = ∑

iγ′i(t)∑

kΓk

ij(t)∂k ,

page 71

2016–17 · Differential Geometry Geodesics · VI.2

where Γkij(t) := Γk

ij(γ(t)

), and therefore

∇γ′X = ∑j∇γ′

(Xj(t)∂j

)= ∑

j

(X′j(t)∂j + Xj(t)∇γ′∂j

)= ∑

k

(X′k(t) + ∑

i,jXj(t)γ′i(t)Γ

kij(t)

)∂k .

(VI.4)

Thus, X is parallel along γ if and only if

X′k(t) + ∑i,j

Xj(t)γ′i(t)Γkij(t) = 0 for all k ∈ u, v. (VI.5)

This is a system of two linear first order differential equations whence there is a unique solutionsatisfying the initial condition X(t0) = X0 (which are two equations as we are in Tγ(t0)

Σ).

VI.2 Geodesics

Geodesics are a special and very important class of curves. They compare to straight lines in R2

in the following ways:

straight lines (of constant speed) are exactly those curves which have constant tangentvector fields and these are parallel, cf. item i) of Example VI.3

straight lines constitute the shortest connections between two points

We will use the first item as a definition for geodesics, then show their existence and (local)uniqueness and finally will address shortest connections in Theorem VI.12.

Definition VI.5 (Geodesics). Let γ : I −→ Σ be a regular parameterised curve. γ is said to begeodesic, if γ′ is parallel along γ, i.e., if ∇dt γ′ = 0 on I.

But what is a good intuition for geodesics? Since ∇dt γ′ = 0, the velocity of the particle γ(t)does not change along its trajectory. This is the case for particles on which no forces whatsoveract or for light rays. (You can also encode the action of forces in the curvature of the surface,leading to curved space-times.)

Example VI.6 (Geodesics).i) In R2 we have ∇dt γ = γ′′, which shows that γ : I −→ R2 is geodesic if and only if γ′′(t) = 0 for

all t. Thus, geodesics in R2 are exactly straight lines of constant speed.

ii) In Examples V.10 and VI.3, we considered two different families of curves on the torus, givenby fixing one of the parameters of the parameterisation: The curves γu(t) = σ(t, v0), runningthrough the hole of the torus, are geodesic for any choice of v0 ∈ R. The curves γv(t) = σ(u0, t)however, which run horizontally around the hole, are geodesic if and only if u0 ∈ πZ.

iii) On S2, the shortest connection between two points p and q is given by the shorter arc of the greatcircle running through these points. A great circle is the intersection of S2 with a plane throughthe origin and it is therefore uniquely determined by giving two distinct points on it. One of

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the exercises shows that a great circle, traversed at constant speed, is a geodesic. Let us notewithout proof that the converse is true as well: The traces of geodesics on S2 are exactly arcs ofgreat circles.

Remark VI.7 (Geodesics and Isometries). Definition VI.5 only involves the curve at hand and the covariantderivative. The latter is preserved under isometries, as we have shown in Corollary V.15. Thus, iff : Σ −→ Σ is an isometry and γ geodesic in Σ, then f∗γ = f γ is geodesic in Σ. In particular, ifσ : U −→ V is a parameterisation of a surface patch V so that I = id, then any geodesic in V is theimage under σ of a line segment of constant speed, cf. Example VI.6, item i).

Theorem VI.8 (Existence and Uniqueness of Geodesics). Let Σ be a surface, p ∈ Σ andX ∈ TpΣ. Then, there is a unique maximal geodesic γp,X : (a, b) −→ Σ such that γ(0) = p andγ′(0) = X.

(Here, −∞ ≤ a < 0 < b ≤ ∞ and “unique maximal” means the following: If γ is a secondgeodesic satisfying these conditions, then tr(γ) ⊂ tr(γp,X).)

Proof : We start by proving local existence and uniqueness and use an approach very muchsimilar to the proof of Proposition VI.4. Thus, let σ : U −→ V be a parameterisation near pand for γ : (−1, 1) −→ V write γ(t) = σ

(u(t), v(t)

)and γ′(t) = γ′u(t)∂u + γ′v(t)∂v. Using

X(t) = γ′(t) in (VI.5), we obtain

∇dt γ′ = 0 ⇐⇒ γ′′k (t) + ∑

i,jγ′i(t)γ

′j(t)Γ

kij(γ(t)) = 0 for all k, (VI.6)

where γ′′k (t) =ddt γ′k(t). This is a system of two nonlinear second order differential equations for

the components of γ and given initial conditions γ(0) = p, γ′(0) = X0, there is a unique solutionγ1

p,X0: (−1, 1) −→ V, by Theorem I.19. (Observe that we have and need four initial conditions:

As we are working in U and Tγ(0)Σ, each condition is “2-dimensional”.)

Then, using γ1p,X0

(± 910 ) and (γ1

p,X0)′(± 9

10 ) to obtain another set of initial conditions near theendpoints of tr(γ1

p,X0), we obtain further geodesics, effectively extending the interval (−1, 1).

Repeating this procedure ad infinitum, we arrive at the unique maximal geodesic γp,X0 .

The system of equations on the right-hand side of (VI.6) are called the geodesic equations. Forinstance, they show that if Γk

ij = 0, a curve is geodesic if and only if γ′′k (t) = 0. For the plane,this means that the geodesics are exactly the straight lines of constant speed.

If, for any choice of p ∈ Σ, X ∈ TpΣ, the maximal geodesic γp,X from Theorem VI.8 is definedon all of R (it exists for all times, so to say), the surface Σ is called (geodesically) complete.

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Example VI.9 (Geodesics on the Cylinder). Let us consider one specific example of geodesics more closely,geodesics on the unit cylinder C. Since C is invariant under rotations in the x-y-plane and translationsin the z-direction, it does not matter, at which explicit point p we look for geodesics. We take forinstance p = (1, 0, 0). Again using the parameterisation σ(u, v) =

(cos u, sin u, v

), we have p = σ(0, 0)

and

TpC = span∂u, ∂v = span

0

10

,

001

.

What is the geodesic running through p in direction of λu∂u + λv∂v, for λu, λv ∈ R? As an ansatz, wedefine the following curve:

γλu ,λv (t) = γ(t) = σ(0 + λut, 0 + λvt)

Then, γ(0) = p and γ′(0) = λu∂u + λv∂v. Moreover,

∇R3

γ′(t)γ′(t) = γ′′(t) = −λ2

u

cos λutsin λut

0

.

Noting that a Gauss map for C near p is given by N = ∂u × ∂v =(

cos u, sin u, 0)t, we see that

∇R3

γ′(t)γ′(t) = −λ2

u N(γ(t)

)and consequently ∇dt γ′(t) = 0. Thus, γ is the geodesic we were looking

for!But why did this turn out so well? Our ansatz was to take a geodesic in R2, namely t 7→ (λut, λvt),

and transport it to C using the parameterisation σ. In Example IV.2, item ii), we have seen that I = id,thus σ is an isometry. But then, any geodesic in C is the image of a line segment as is γ.

As geodesics are intrinsic, there also is an alternative form of the geodesic equations (VI.6)using only the local components of the first fundamental form. Using the fact that a curve γ isa geodesic if and only if γ′′ is orthogonal to both ∂u and ∂v, we obtain:

Proposition VI.10. With notation as in the proof of Theorem VI.8 and with E(u(t), v(t),F(u(t), v(t)), G(u(t), v(t)) denoting the local components of I, the geodesic equations (VI.6)are equivalent to:

ddt

(γ′uE + γ′vF

)=

12

((γ′u)

2 ∂E∂u

+ 2γ′uγ′v∂F∂u

+ (γ′v)2 ∂G

∂u

)ddt

(γ′uF + γ′vG

)=

12

((γ′u)

2 ∂E∂v

+ 2γ′uγ′v∂F∂v

+ (γ′v)2 ∂G

∂v

) (VI.7)

Geodesics and Shortest Connections

Given two points p, p′ in a surface Σ, we write γ : p p′, if γ is a curve in Σ starting at p andending at p′. Suppose that for any choice of p, p′, there is at least one curve γ : p p′. (Then,

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Σ is called path-connected.) We define the distance from p to p′ to be

dist(p, p′) := inf`(γ)

∣∣ γ : p p′

. (VI.8)

It is a good exercise to show that this turns Σ into a metric space. Now let γ : [0, `] −→ Σbe a regular curve of unit speed so that γ(0) = p, γ(`) = p′. We want to study how arclength changes as we vary the curve γ slightly. For this, we let a variation of γ be a C2 functionh : [0, `]× (−ε, ε) −→ Σ so that

h(s, 0) = γ(s) , h(0, t) = p , h(`, t) = p′ for all s and t, respectively.

Thus, h(s, t) =: γt(s) : p p′ is a family of curves connecting p and p′ and where γ0(s) is unitspeed. The arc length of any member of this family is given by

L(t) :=`∫

0

∣∣∣∂h∂s

(s, t)∣∣∣ ds =

`∫0

∣∣γ′t(s)∣∣ ds (VI.9)

and we may compute its derivative at t = 0:

Proposition VI.11 (The First Variation of Arc Length). With the same notation as in thepreceding paragraph, the function L : (−ε, ε) −→ R is continuously differentiable at t = 0 andits derivative is given by

L′(0) = −`∫

0

⟨∇ds

γ′(s) ,ddt

∣∣∣t=0

γt(s)⟩

ds .

This is called the first variation of arc length.

Proof : Using the fact that γ = h(·, 0) is parameterised by arc length and hence∣∣ ∂h

∂s (s, t)∣∣ 6= 0

in a neighbourhood of t = 0, it is not difficult to see that L is continuously differentiable neart = 0 and that its derivative is obtained by differentiating under the integral. Before we startcalculating L′(0), note that ∂2h

∂s∂t =∂2h∂t∂s and we obtain

∇ ∂h∂s

∂h∂t

= ∇ ∂h∂t

∂h∂s

where we view ∂h∂s and ∂h

∂t as vector fields on Σ (cf. (V.7)). But then,

L′(t) =`∫

0

12

∣∣∣∣∂h∂s

(s, t)∣∣∣∣−1 d

dt

(⟨∂h∂s

(s, t) ,∂h∂s

(s, t)⟩)

ds

=

`∫0

∣∣∣∣∂h∂s

(s, t)∣∣∣∣−1 ⟨∇

dt∂h∂s

(s, t) ,∂h∂s

(s, t)⟩

ds

=

`∫0

∣∣∣∣∂h∂s

(s, t)∣∣∣∣−1 ⟨∇

ds∂h∂t

(s, t) ,∂h∂s

(s, t)⟩

ds

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2016–17 · Differential Geometry Normal and Geodesic Curvature · VI.3

and evaluation at t = 0 yields

L′(0) =`∫

0

−⟨

∂h∂t

(s, 0) ,∇ds

∂h∂s

(s, 0)⟩+

dds

⟨∂h∂t

(s, 0) ,∂h∂s

(s, 0)⟩

ds

= −`∫

0

⟨∂h∂t

(s, 0) ,∇ds

∂h∂s

(s, 0)⟩

ds +⟨

∂h∂t

(s, 0) ,∂h∂s

(s, 0)⟩ ∣∣∣∣`

s=0

= −`∫

0

⟨ ddt γ0(s) , ∇ds γ′0(s)

⟩ds ,

which proves the claim.

Using this formula, we can study the relation between shortest connections and geodesics,though we will not give a proof. The key point is showing that a curve γ is a geodesic if and onlyif we have L′h(0) = 0 for any chosen variation h of γ. (We used a subscript h here to emphasisethe dependence on h.)

Theorem VI.12. Let Σ be a connected surface.

i) If p, p′ ∈ Σ and γ : p p′ satisfies `(γ) = dist(p, p′), then γ is a geodesic.

ii) For any p0 ∈ Σ, there is an open neighbourhood V ⊂ Σ of p0 so that the following holds:For any p, p′ ∈ Σ there is a unique geodesic γ : p p′ and this geodesic satisfies`(γ) = dist(p, p′).

We have already seen that item ii) cannot be true globally: The inner equator of the torus is aclosed geodesic, see Example VI.6, item ii) and taking any two points on the inner equator whichare not antipodal, we have two geodesics of different lengths connecting these two points.

VI.3 Normal and Geodesic Curvature

In Chapter II we have studied the curvature of curves in R3, in Chapter IV that of surfaces in R3.We now want to combine these two approaches. If Σ is a surface, p ∈ Σ and X ∈ TpΣ is of unitlength, let γ : I → Σ be a regular parameterised curve such that γ(t0) = p and γ′(t0) = X. Ifγ was parameterised by arc length, then γ′′(t0) would give the curvature of γ, but this dependson the particular choice of curve γ representing X. As the second fundamental form (which isused to define curvatures for Σ) does not depend on the particular parameterisation of Σ, this isan unsatisfactory situation. It turns out though, that the normal part of γ′′(t0) behaves in a betterway.

Definition VI.13 (Normal Curvature). Let Σ be an oriented surface, γ : I −→ Σ be regular andunit speed and γ(s0) = p ∈ Σ. Then,

⟨γ(s0) , N(p)

⟩is called the normal curvature of γ in Σ at p

and denoted by κn(γ, p).

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Even though the vector γ(s0) does depend on the choice of γ, the normal curvature does sur-prisingly only depend on X = γ(s0) (and of course Σ, p and N):

Proposition VI.14. In the situation of Definition VI.13, the normal curvature of γ in Σ at pdepends only on Σ, p and X = γ(s0). We therefore also call it the normal curvature of Σ at pin direction X and sometimes denote it by κn(X). Moreover, we have

〈γ(s0) , N(p)〉 = IIp(X) .

Proof : Since N(γ(s)) ⊥ γ(s), the trick of differentiating their scalar product with respect to sand evaluating at s = s0 yields

0 =dds

∣∣∣s=s0

⟨γ(s) , N(p)

⟩=⟨γ(s) , N(γ(s))

⟩∣∣∣s=s0

+⟨γ(s) , d

ds N(γ(s))⟩∣∣∣

s=s0

=⟨γ(s0) , N(γ(s0))

⟩+⟨γ(s0) ,−W

∣∣γ(s0)

(γ(s0))⟩

,

and thus κn(γ(s0)) =⟨

X , W∣∣

p(X)⟩= IIp(X), where p = γ(s0) and X = γ(s0).

Now we proceed as in the case of curves and define a second quantity measuring curvature:We extend the pair γ, N of linearly independent unit vectors to a basis of R3 by letting

n(p) := N(p)× γ(s) , (VI.10)

where again p = γ(s). Then, γ(s), n(p), N(p) is a right-handed orthonormal frame for anyp ∈ tr(γ). Since γ ⊥ γ, there is a function κg(p) so that

γ(s) = κn(γ(s)

)N(p) + κg(p)n(p) .

The vector κn(γ(s)

)N(p) is the normal part of ∇R3

γ γ(s) and similarly, κg(p)n(p) is its tangentialpart. Note that κg can hence be given by

κg(p) =⟨∇

ds γ(s) , n(p)⟩=⟨∇

ds γ(s) , N(p)× γ(s)⟩

. (VI.11)

Definition VI.15 (Geodesic Curvature). The function

κg(γ, p) :=⟨∇

ds γ(s) , N(p)× γ(s)⟩

is called the geodesic curvature of γ in Σ at p.

Clearly, a unit speed curve will have vanishing geodesic curvature, κg = 0, if and only if it isa geodesic. Thus, unit speed geodesics are curves whose curvature stems from the curvature ofΣ only. This also shows that geodesic curvatures does depend on the choice of parameterisation:Reparametrising a geodesic to have non-constant speed will turn it into a “non-geodesic” withnonvanishing geodesic curvature. (As an exercise, it will be shown that a curve is a geodesic ifand only if it is of constant speed and has vanishing geodesic curvature.)

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Remark VI.16 (Relations between Curvatures for Curves). Since the curvature κ of the unit speed curve γ

as a curve in R3 is given by |γ|, see Definition II.13, we have

κ2 = κ2g + κ2

n .

This identity becomes clearer if you note which curvature carries which information:

κ describes the curvature of γ in R3

κg describes the curvature of γ in Σ κn describes the curvature of Σ in R3

Please observe that, while κn and κg depend on the orientation N and hence carry a sign, the curvatureκ is independent of the chosen orientation and always positive.

Moreover, note that the frame γ, n, N is in general not the Frenet-Serret frame for γ. In fact,using the Frenet-Serret equations, we see that κNγ = κnNΣ + κgn, using subscripts to distinguish thenormals. Thus, we have Nγ = NΣ if and only if γ is a geodesic. (In which case we then have n = −B.)

page 78

Chapter VII

The Theorem of Gauss-Bonnet

In this last chapter, we would like to give a (noncomplete) proof of the Theorem of Gauss-Bonnet. This surprising result, giving a connection between the Gauss curvature of a surfaceand a topological invariant, has as starting point the elementary fact that the interior angles of atriangle in R2 add up to π. Gauss generalised this to geodesic triangles in surfaces and Bonnetfurther generalised this to geodesic polygons in surfaces. Today, a slightly more general resultis known as the Local Theorem of Gauss-Bonnet. It has seen many generalisations to differentcontexts and could be seen to be at the heart of modern differential geometry and index theory.

VII.1 Chains in Surfaces

As usual, let Σ be a surface and σ : U −→ V ⊂ Σ be a local parameterisation. We may definesimple closed curves in Σ very similarly to those in R2, but have to be careful in one importantaspect: Theorem II.23 ensures that the interior of a curve is a well-defined bounded set and wewould like to have this in the case of simple closed curves in surfaces as well. But this willgenerally not hold, as you can see by looking at the examples of the unit sphere S2 ⊂ R3 andthe unit cylinder C ⊂ R3: In both cases, the equator is the trace of what we might call a simpleclosed curve, but in the case of the sphere, the surface is cut into two bounded open sets, while inthe case of the cylinder, it is cut into two unbounded open sets.

Though we will not be able to avoid the first situation (in which case the interior will dependon the chosen parameterisation), but we definitely want to exclude the second situation.

Definition VII.1 (Simple Closed Curves, Chains and Polygons in Surfaces).

i) A parameterised curve γ : [a, b] −→ Σ is called a simple closed curve in Σ if there is a localparameterisation σ : U −→ V and a simple closed curve γ : [a, b] −→ U so that

γ = σ γ and int(γ) ⊂ U .

In this case, we call int(γ) := σ(int(γ)

)the interior of γ.

ii) A continuous parameterised curve γ : [a, b] −→ Σ is called a chain in Σ if there is a paramet-erisation σ : U −→ V and a plane chain γ : [a, b] −→ U so that

γ = σ γ and int(γ) ⊂ U .

In this case, we call σ(int(γ)

)the interior of γ.

iii) If P ⊂ Σ and there is a chain γ so that P = int(γ), we call P a (generalised) polygon. Theedges and vertices of P are exactly the arcs and vertices of γ, compare Definition II.29.

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2016–17 · Differential Geometry The Local Theorem of Gauss-Bonnet · VII.2

Generalising the idea of the interior of the curve of lying on the left side, we can define positiveorientations for chains in surfaces:

Definition VII.2. Let γ : [a, b] −→ Σ be a chain in an oriented surface Σ and let N be the respectiveGauss map for Σ. We say that γ is oriented positively (with respect to N) if and only if int(γ) lieson the left side of the oriented plane given by the frame N, γ′. Equivalently, if and only if N × γ′

always points into int(γ).

That this is indeed a direct generalisation of Definition II.24 can be seen by endowing R2 ⊂ R3

with the orientation given by N = (0, 0, 1) and comparing the two definitions. (This choice of Nlinks the two “usual” orientations of R2 and R3.)

Now, given any orthonormal frame e1, e2 along γ, we may also define angular functions forγ with respect to e1, e2 exactly as before in Definition II.26: α is assumed to be at least C1 awayfrom the vertices of γ and supposed to satisfy

γ′(t)|γ′(t)| = cos

(α(t)

)e1(γ(t)

)+ sin

(α(t)

)e2(γ(t)

)as long γ(t) is not a vertex. The exterior angles are again the differences of limits of the angularfunction when approaching a point from opposite sides, chosen in [−π, π] (cf. Definition II.30).

VII.2 The Local Theorem of Gauss-Bonnet

The analogues of the Theorems of Turning Tangents, Theorems II.27 and II.31, contained anintegral over the derivative of an angular function. The disadvantage of this is that angularfunctions depend on the choice of a frame. But there is an invariant way of formulating these(and at the same time generalising these to surfaces) by using the geodesic curvature. In thisprocess, a correction term involving the Gauss curvature arises.

Theorem VII.3 (of Gauss-Bonnet – local, smooth). Let γ : [0, `] −→ Σ be a regular simpleclosed curve, parameterised by arc length and positively oriented. Then,∫

γ

κg d` +∫

int(γ)

K dS = 2π .

Let us put the technicalities of the proof of Theorem VII.3 into two lemmata. First of all,note that the integral involving the angular function α was replaced by one over the geodesiccurvature. This is desirable as angular functions depend on choices of frames while the geodesiccurvature does not.

Lemma VII.4. Let Σ be an orientable surface and γ be as in Theorem VII.3. Let e1, e2 ⊂ X(γ) bean orthonormal frame along γ and let α be an angular function for γ with respect to this frame. Then,with respect to the Gauss map N = e1 × e2,

κg = α−⟨e1 , ∇ds e2

⟩.

page 80


Proof : Using the right-handedness of e1, e2 and the properties of the covariant derivative, wecalculate

∇ds γ = ∇

ds e1 cos α + ∇ds e2 sin α +

(e2 cos α− e1 sin α

)α ,

N × γ = e2 cos α− e1 sin α ,

and recall from (VI.11) that κg =⟨∇

ds γ , N× γ⟩. Using the orthonormality of e1, e2 we also have⟨

e1 , ∇ds e1⟩=⟨e2 , ∇ds e2

⟩= 0 and

⟨e1 , ∇ds e2

⟩= −

⟨e2 , ∇ds e1

⟩. Together, these equations show that

κg = α−⟨e1 , ∇ds e2

⟩.

Thus, we may exchange the coordinate dependent integral involving an angular function byan integral over the (invariantly defined) geodesic curvature, plus a correction term involving⟨

e1 , ∇ds e2⟩. This latter term does again depend on the choice of a frame e1, e2, but only appar-

ently so: The next lemma will link this term to the (again invariantly defined) Gauss curvature.

Lemma VII.5. Let e1, e2 be an orthonormal frame on a surface patch V and let N = e1× e2 be theGauss map for V. Then,

i) ∂∂u N × ∂

∂v N = K · ∂u × ∂v and

ii) ∂∂u N × ∂

∂v N =(⟨

∂∂u e1 , ∂

∂v e2⟩−⟨

∂∂v e1 , ∂

∂u e2⟩)

N.

Proof : First of all, we note that, if e = e1, e2, e3 is any choice of frame for R3 on V, we mayidentify vectors a ∈ R3 with their coefficient vectors a with respect to e. Then, if we let e bethe vector with symbols ej as entries and if we identify λ · ej ≡ λ · ej for λ ∈ R, we may writea× b = det

(a, b, e

), for any a, b ∈ R3. Omitting the tildes for simplicity, if K 6= 0, we obtain

∂∂u N × ∂

∂v N = det(

∂∂u N, ∂

∂v N, e)= det

(−W(∂u),−W(∂v), e

)= det W

(∂u, ∂v, W−1e

)= det W · det

(∂u, ∂v, W−1e

)= K · ∂u × ∂v ,

as W will be invertible (for K 6= 0) and W−1e will be a frame as well. If K = 0, we still obtain∂

∂u N× ∂∂v N = det

(W(∂u), W(∂v), e

)and this has to vanish as W has nontrivial kernel, thus both

sides of item i) vanish. This proves the first item. For the second item, we note that ∂∂u N × ∂

∂v Nis a normal vector (or 0) and compute

∂∂u N × ∂

∂v N =((

∂∂u e1 × e2

)+(e1 × ∂

∂u e2))×((

∂∂v e1 × e2

)+(e1 × ∂

∂v e2))

=(⟨

∂∂u e1 , N

⟩N × e2 +

⟨∂

∂u e2 , N⟩

e1 × N)×(⟨

∂∂v e1 , N

⟩N × e2 +

⟨∂

∂v e2 , N⟩

e1 × N)

=(⟨

∂∂u e1 , N

⟩e1 +

⟨∂

∂u e2 , N⟩

e2

)×(⟨

∂∂v e1 , N

⟩e1 +

⟨∂

∂v e2 , N⟩

e2

)=(⟨

∂∂u e1 , N

⟩⟨∂

∂v e2 , N⟩−⟨

∂∂v e1 , N

⟩⟨∂

∂u e2 , N⟩)

N

=(⟨

∂∂u e1 , ∂

∂v e2⟩−⟨

∂∂v e1 , ∂

∂u e2⟩)

N ,

where we have used the fact that e1, e2, N is a right-handed orthonormal frame and, in the lastline, that, e.g., ∂

∂u e1 is perpendicular to e1 and hence a linear combination of e2 and N.

page 81


The final puzzle piece is given by the Theorem of Green (or of Gauss-Green, as it is a specialcase of the Theorem of Gauss). This states that, whenever γ : [0, `] −→ R2 is a simple closed planecurve with V = int(γ) and P, Q : V −→ R are continuously differentiable, then

`∫0

uP + vQ ds =∫V

∂∂u Q− ∂

∂v P dudv , (VII.1)

where u,v both denote coordinates in V and γ(s) =(u(s), v(s)

). Now we are prepared to prove

the smooth version of the local Theorem of Gauss-Bonnet.

Proof (of Theorem VII.3) : To begin with, let us consider the integral

I :=`∫

0

⟨e1(γ(s)

), ∇ds e2

(γ(s)

)⟩ds .

Since e1 is perpendicular to N, we have⟨e1(γ((s)

), ∇ds e2

(γ((s)

)⟩=⟨e1(γ((s)

), d

ds

(e2(γ((s)

))⟩and writing γ(s) = σ

(u(s), v(s)

)we obtain

dds

(e2(γ(s)

))= d

ds

((e2 σ

)(u(s), v(s)

))= De2

(u∂u + v∂v

)=

∂e2

∂uu +

∂e2

∂vv .

But then,

I =

`∫0

⟨e1 , ∂

∂u e2⟩u +

⟨e1 , ∂

∂v e2⟩v ds ,

and using Green’s Theorem, (VII.1), we arrive at

I =∫

int(γ)

∂∂u(⟨

e1 , ∂∂v e2

⟩)− ∂

∂v(⟨

e1 , ∂∂u e2

⟩)dudv

=∫

int(γ)

⟨∂

∂u e1 , ∂∂v e2

⟩−⟨

∂∂v e1 , ∂

∂u e2⟩

dudv .

As e1, e2, N is a right-handed orthonormal frame, using Lemma VII.5 we obtain

I =∫

int(γ)

⟨∂

∂u N × ∂∂v N , N

⟩dudv =

∫int(γ)

K∣∣∂u × ∂v

∣∣ dudv

=∫

int(γ)

K dS .

On the other hand, using Lemma VII.4, we see that

I =

`∫0

α(s)− κg(s) ds .

Since α is an angular function for γ with respect to e1, e2, it will also be an angular functionfor γ := σ−1 γ with respect to Dσ−1e1, Dσ−1e2. Then, by Theorem II.27,

2π −`∫

0

κg(s) ds = I =∫

int(γ)

K dS ,

which completes the proof.

page 82


Recalling how we obtained Theorem II.31 from Theorem II.27 (by inscribing a family ofsmooth curves and pushing it out into the corners), we may generalise Theorem VII.3 in exactlythe same way: We stay local (for the moment), but include corners again. (We will not repeatthe actual proof, though.)

Theorem VII.6 (of Gauss-Bonnet, local). Let P ⊂ Σ be a generalised polygon with exteriorangles ϑ1, . . . , ϑn. Let K be the Gauss curvature of Σ and κg be the geodesic curvature of apositively oriented unit speed chain γ so that P = int(γ). Then,∫

∂P

κg d`+∫P

K dS +n

∑i=1

ϑi = 2π .

Remark VII.7 (Local Gauss-Bonnet and Orientations). As we have mentioned at the end of Section VII.1,for a chain γ : [0, `] −→ Σ to be positively oriented depends on the choice of Gauss map for Σ. Itis positively oriented with respect to N if and only if it is negatively oriented with respect to −N.Changing the direction of γ also changes the orientation: If γ(s) = γ(` − s), then γ is positivelyoriented if and only if γ is negatively oriented (i.e. not positively oriented). Now, if κg(γ, s) denotesthe geodesic curvature of γ at γ(s), we see that κg(γ, s) = −κg(γ, `− s) and so

∫γ

κg(γ) d` =`∫

0

κg(γ, s) ds = −`∫

0

κg(γ, `− s) ds =0∫`

κg(γ, s) ds = −∫γ

κg(γ) d` .

Similarly, the exterior angles change their sign: ϑi(γ) = −ϑn−i(γ). On the other hand, both the Gausscurvature and the surface integral are independent of orientations. Consequently, fixing a Gauss mapN for Σ, we obtain∫

γ

κg(γ) d`+n

∑i=1

ϑi(γ) = ±(

2π −∫

int(γ)

K dS)

with the sign depending on the orientation of γ. For Theorem VII.6 to hold in the above formulation,we indeed do need γ to parameterise ∂P positively! Then again, Theorem VII.6 does not depend onthe chosen Gauss map N (or orientation) for Σ: As long as we take the geodesic curvature and exteriorangles of a positive parameterisation, the formula will hold.

Example VII.8 (on the Local Gauss-Bonnet).i) For (true) polygons in R2, we reobtain formula (II.14): As κg = 0 and K = 0, we are left with

n

∑i=1

ϑi = 2π .

ii) If P is a geodesic triangle in the unit sphere S2, Theorem VII.6 shows that

3

∑i=1

ϑi = 2π −∫P

K dS = 2π − area(P) ,

page 83

2016–17 · Differential Geometry The Global Theorem of Gauss-Bonnet · VII.3

which shows that for the interior angles φi = π − ϑi we have ∑i φi = π + area(P). Thus, thesum of the angles in a triangle on the sphere depends on the size of the triangle! More generally,we see that the interior angles of geodesic triangles in surfaces with Gauss curvature K satisfy:

∑i

φi > π if K > 0

∑i

φi = π if K = 0

∑i

φi < π if K < 0

iii) If γ : [0, `] −→ Σ is a simple closed curve and a unit speed geodesic with interior P, then κg = 0and all ϑi = 0. Therefore,∫

P

K dS = 2π .

This is insofar a very interesting formula, as we have a direct link between area and curvature ofa surface patch! But beware, the prerequisite that P is the interior of a simple closed geodesic isimportant. (For instance, there are no simple closed geodesics in R2 or in the cylinder C.) Thiswill be generalised in the global version of the Theorem of Gauss-Bonnet.

VII.3 The Global Theorem of Gauss-Bonnet

The Local Theorem of Gauss-Bonnet, Theorem VII.6, is valid for generalised polygons in sur-faces, i.e., for the interior set of chains in surfaces. In order to turn global, we consider unions ofsuch generalised polygons. But we will only consider a very special type of union, which is onthe one hand almost disjoint and on the other hand respects the structure of polygons:

Definition VII.9 (Polygonal Covers, Euler Characteristic). Let Σ be a surface and V ⊂ Σ becompact. A polygonal cover of V is a set P =

P1, . . . , Pm

, where:

i) Each Pi is a generalised polygon in Σ, i.e., there is a chain γi in Σ such that Pi = int(γi),

ii) The closures of the Pi cover V: V =m⋃

i=1Pi,

iii) The intersections Pi ∩ Pj, for i 6= j, are either empty, a common vertex or a common edge.

We denote by f (P) the number of faces (polygons) of P , by e(P) the number of edges of P and byv(P) the number of vertices of P . (Each edge and vertex is counted only once even though it mightpertain to multiple polygons.) The Euler Characteristic of P is defined as

χ(P) := f (P)− e(P) + v(P) .

If we take a polygonal cover P and cut its members into smaller polygons, the result iscalled a subdivision or refinement and again a polygonal cover. We can, for instance, always finda subdivision which consists of triangles only, such polygonal covers are called triangulations.We now state three important facts about polygonal covers and the Euler characteristic, thoughwithout proof:

page 84


Proposition VII.10. Let V be a compact subset of a surface Σ and suppose ∂V is piecewisesmooth.

i) There exists a finite triangulation of V.

ii) The Euler characteristics of any two polygonal covers of V are the same.

iii) If there is a homeomorphism ϕ : V −→ V′, then χ(V) = χ(V′).

Note that items i) and ii) allow us to define the Euler characteristic χ(V) for any compactsubset of a surface as long as it has piecewise smooth boundary. Thus, the Euler characteristicis some sort of combinatorial invariant of a compact subset V ⊂ Σ and as such is independent ofactual parameterisations or orientations (and even orientability!).

But item iii) states that it is at the same time a topological invariant. The combination of thesetwo renders it very useful: We have the computability of a combinatorial object paired with thestability of a topological object.

Example VII.11 (On polygonal covers and the Euler Characteristic).i) Imagine a tetrahedron T inside of a sphere. Projecting radially outwards onto the sphere gives a

continuous mapping T −→ S2. This is in fact a homeomorphism and leads to a polygonal coverof the sphere with 4 faces, 6 edges and 4 vertices. This shows that

χ(S2) = 4− 6 + 4 = 2 .

ii) Take a square and identify opposing sides. This is one way of constructing the torus, though youwill still need to find an embedding into R2 to call it a surface. To obtain the Euler characteristicof the torus, we need to count the number of faces, edges and vertices after identifying theopposing sides. This results in having 1 face, 2 edges and 1 vertex. Hence

χ(torus

)= 1− 2 + 1 = 0 .

iii) Now consider a compact set V ⊂ R2 and let P be a triangulation of V. If P ∈ P is an interiortriangle, i.e., if P ∩ ∂V = ∅, then removing P from P will leave e(P) and v(P) unchanged butreduce f (P) by 1. On the other hand, removing a single triangle amounts to punching a hole inV. Thus, if D ⊂ V is a disc,

χ(V \ D

)= χ

(V)− 1 .

iv) The converse of item iii) of Proposition VII.10 does not hold: Consider a rectangular strip givenas(x, y) ∈ R2

∣∣ 0 ≤ x ≤ 2 , 0 ≤ y ≤ 1

and divide it into four equilateral triangles. Thenidentify the sides x = 0 and x = 2. The result will have Euler characteristic

χ = 4− 8 + 4 = 0 .

But there are (at least) two ways in which we can identify these edges: By respecting the orient-ation or by reversing the orientation, i.e., giving the strip half a twist before gluing. The first willresult in a piece of a cylinder while the second will give a Möbius strip. As the Möbius strip isnonorientable, these two are clearly not homeomorphic!

page 85


Remark VII.12 (On the Triangulation Conjecture). The Triangulation Conjecture states the following:

Every compact topological manifold M possesses a locally finite triangulation P .

Here, locally finite means that each point p ∈ M has a neighbourhood which intersects only finitelymany members of P (in general, triangulations need not be finite themselves). Proposition VII.10, itemi), is the Triangulation Conjecture for differentiable, 2–dimensional manifolds which are embedded inR3. The Triangulation Conjecture is known to be true in dimensions 2 and 3 since the 1920s and 1950s,respectively. But, in 2013, Manolescu33 filled in the missing part in a proof that the TriangulationConjecture is wrong in any dimension greater or equal to 4.

With these preparations in place, we are now in the position to state and prove the globalversion of the Theorem of Gauss-Bonnet.

Theorem VII.13 (of Gauss-Bonnet, global). Let Σ be a surface and V ⊂ Σ be compact andhave piecewise smooth boundary with exterior angles ϑ1, . . . , ϑn. Let K be the Gauss curvature ofΣ and κg be the geodesic curvature of a curve parameterising ∂V. Then,∫

∂V

κg d`+∫V

K dS +n

∑i=1

ϑi = 2π χ(V)

.

Remark VII.14 (On Orientations in Polygonal Covers). Before we start with the proof of Theorem VII.13,note the following: Let V ⊂ Σ be compact, P =

P1, . . . , Pm

be a polygonal cover for V and suppose

that γ : [0, `] −→ Σ parameterises ∂V positively and by arc length. (That is, we fix a Gauss map Nfor Σ and assume that γ is oriented positively with respect to N.) N of course defines Gauss maps foreach polygon Pj as well and we may choose positively oriented unit speed parameterisations γj of ∂Pj.Then, if E is any interior edge in P , E is parameterised by (parts of) exactly two curves γj1 and γj2 asE is adjacent to exactly two polygons Pj1 and Pj2 . But then, since the γj’s are all positively oriented, γj1and γj2 have to run along E in opposite directions! (As Pj1 and Pj2 are on different sides of E.)

Proof : Let P =

P1, . . . , Pm be a polygonal cover of V. We start by using the local versionTheorem VII.6 for each Pj ∈ P and then taking the sum over all j:

m

∑j=1

2π =m

∑j=1

∫∂Pj

κjg d`+

m

∑j=1

∫Pj

K dS +m

∑j=1

v(Pj)

∑i=1

ϑji ,

where κjg denotes the geodesic curvature of a chain parameterising ∂Pj and the ϑ

ji are the exterior

angles of Pj.

33 Ciprian Manolescu, ∗ 1978 in Alexandria, Romania

page 86


When choosing the chains γj parameterising the ∂Pj, we need to take care that they have thecorrect orientation. Whenever two polygons P, P′ have a common edge, the parameterisations γ

and γ′ need to run along this edge in different directions. This is always possible to achieve andwill result in ∂V being run along in a single direction.

Now let us consider each sum independently:

As m = f (P), we have ∑mj=1 2π = 2π f (P).

The Pj cover V except for a set of measure 0, and so

m

∑j=1

∫Pj

K dS =∫V

K dS .

For the geodesic curvatures, we split the sum further:

m

∑j=1

∫∂Pj

κjg d` =

m

∑j=1

e(Pj)

∑i=1

∫Ej

i

κg(Eji ) d` ,

where the Eji denote the edges of Pj. Now if Ej

i is an interior edge in P , we will integrateover this edge twice, each time with respect to a different orientation and hence theseintegrals cancel. If it is an exterior edge, we will just integrate once, resulting in

m

∑j=1

∫∂Pj

κjg d` =

∫∂V

κg d` .

For the final sum, the one over the exterior angles, let us consider a single polygon Pj first.

As ϑji = π − φ

ji for the interior angles φ

ji , we have

v(Pj)

∑i=1

ϑji =

v(Pj)

∑i=1

π − φji = π v(Pj)−

v(Pj)

∑i=1

φji .

The next step is to sum over all polygons. Note that v(Pj) = e(Pj), so the first of the latterterms results in

m

∑i=1

π v(Pj) = 2π e(P)− πn ,

as each edge is counted twice apart from the n exterior edges. The sum over the interiorangles gives us a summand 2π whenever we add up all the angles for an interior vertex,and the full interior angle of V at an exterior vertex. So,

m

∑j=1

v(Pj)

∑i=1

φji = 2π

(v(P)− n

)+

n

∑k=1

φk = 2π v(P)− πn−n

∑k=1

ϑk .

But then, these combine to give

m

∑j=1

v(Pj)

∑i=1

ϑji = 2π e(P)− πn− 2π v(P) + πn +

n

∑k=1

ϑk = 2π(e(P)− v(P)

)+

n

∑i=1

ϑi .

page 87

2016–17 · Differential Geometry Applications of Gauss-Bonnet · VII.4

Combining above four calculations, we arrive at∫∂V

κg d`+∫V

K dS +n

∑i=1

ϑi = 2π(

f (P)− e(P) + v(P))= 2π χ(P) = 2π χ(V) .

VII.4 Applications of Gauss-Bonnet

The Theorem of Gauss-Bonnet, Theorem VII.13 has some interesting generalisations and con-sequences, we only present three here.

Corollary VII.15 (Gauss-Bonnet for Compact Surfaces). Let Σ be a compact surface in R3.Then, ∫

Σ

K dS = 2π χ(Σ) .

Though this is just a special case of Theorem VII.13, in the absence of boundary, it is a veryinteresting formula. Consider for instance the following: The right-hand side 2πχ(Σ) will notvary under small perturbations of Σ, it is (as claimed in Proposition VII.10) an invariant forhomeomorphy. Thus, if we distort the surface, changing it and its curvature, as long as we donot place cuts or glue parts in, the total curvature

∫K dS does not change!

Using a direct, constructive approach (which we will not describe here), Corollary VII.15

leads to a complete classification of compact, connected and orientable surfaces: If Σ is such asurface, we define

g(Σ) := 12(2− χ(Σ)

)(VII.2)

and call this the genus of Σ. Then:

If Σ is a compact, connected and orientable surface, then g(Σ) ∈ N0. Moreover, twosuch surfaces are homeomorphic if and only if they are of the same genus (or, equival-ently, they have the same Euler characteristic).

(VII.3)

The genus measures the number of holes in Σ, for instance the sphere has genus 0 while the torushas genus 1. It turns out that every compact, connected and orientable surface is a connected sumof tori, i.e., a number of tori which are glued together, and g(Σ) tells us how many tori we need.Note that (VII.3) is one of the nicest classification results, though. If we remove the prerequisiteof compactness or increase the dimension, for instance, the story will be more complicated (oreven unsolved).

Corollary VII.16. Let Σ be a compact, connected and oriented surface with nonnegative Gausscurvature K which is not identically 0. Then, Σ is homeomorphic to S2.

page 88


This is in fact a pretty strong result: We know that any sphere has positive constant Gausscurvature. But Corollary VII.16 tells us that Σ essentially is a sphere as soon as it has nowherenegative and somewhere positive curvature! The proof uses the classification (VII.3) and Co-rollary VII.15: Any compact, connected and orientable surface Σ satisfying χ(Σ) > 0 is homeo-morphic to the sphere (as χ(Σ) > 0 only for g(Σ) = 0) and K ≥ 0 yet K 6= 0 implies χ(Σ) > 0.

The last application we will present is one of the rare theorems which are easy to describe toa lay audience but nevertheless not trivial to prove:

Corollary VII.17 (Hairy Ball Theorem). You cannot comb a hairy ball. More precisely, ifX ∈ X(S2), then there is p ∈ S2 such that X(p) = 0.

This is in fact a corollary to the following much more general result:

Corollary VII.18 (Poincaré-Hopf Theorem). Let Σ be a compact, oriented surface and letX ∈ X(Σ) have only finitely many stationary points p1, . . . , pn. Then,

n

∑i=1

µX(pi) = χ(Σ) ,

where µX(pi) denotes the multiplicity of pi in X.

Here, we need to explain a few things. A stationary point of a vector field is a point p ∈ Σsuch that X(p) = 0. This comes from the visualisation that a vector field describes a flow (forinstance of a fluid) on the surface, and the flow is stationary exactly at a zero of the vector field.The multiplicity of a stationary point describes the way the vector field winds around the pointand is hence related to angular functions.

Now let V be a surface patch containing the stationary point p and e1, e2 be an orthonormalframe on V. Choose a positively oriented, simple closed curve of unit speed γ in V whichcontains p in its interior, p ∈ int(γ) ⊂ V, but so that p is the only zero of X in int(γ). Let α bean angular function for X

|X| along γ. Then we define

µ(p) := µX(p) :=1

2π

`(γ)∫0

α(s) ds .

The curve X|X| (s) := X

|X|(γ(s)

)is, by assumption, regular and it is also closed. Compare this to

the Theorem of Hopf, Theorem II.27: If X|X| = γ, then µ(p) = 1. But X

|X| need not be simple. Inany case, similar arguments to those leading to Theorem II.27 show that µ(p) ∈ Z. Moreover,the multiplicity does not depend on the choices we have made in order to define it, i.e., the framee1, e2, the curve γ and the angular function α. Visually, µ(p) counts how often the curve X

|X| (s)runs around the origin.

page 89


Proof (of Corollary VII.18) : Let γ1, . . . , γn be positively oriented regular simple closed curvesof unit speed in Σ with interior sets Vi := int(γi). Assume that pi ∈ Vi and that Vi ∩Vj = ∅ fori 6= j. Then let

Σ′ := Σ \(

n⋃i=1

Vi

).

Using Theorem VII.13 and defining V :=⋃

i Vi,∫Σ′

K dS +∫V

K dS = 2π χ(Σ) ,

and using calculations similar to those which lead to the proof of the Local Theorem of Gauss-Bonnet, cf. Lemma VII.5 and Theorem VII.3, we see that∫

Σ′

K dS = −n

∑i=1

∫∂Vi

⟨e1 , ∇ds e2

⟩d`

and ∫V

K dS =n

∑i=1

∫∂Vi

⟨f1 , ∇ds f2

⟩d` ,

where e1, e2 and f1, f2 are orthonormal frames over Σ′ respectively V. (Please observe thedifference in signs. This is due to ∂Vi being positively oriented for Vi and hence negativelyoriented for Σ′.) But then,

n

∑i=1

∫∂Vi

⟨f1 , ∇ds f2

⟩−⟨e1 , ∇ds e2

⟩d` = 2πχ(Σ) .

On the other hand, following Lemma VII.4, we have⟨e1 , ∇dt e2

⟩∣∣∂Vi

= αi − κig and

⟨f1 , ∇ds f2

⟩∣∣∂Vi

= βi − κig ,

where αi, βi are angular functions for γi with respect to the frames e1, e2 respectively f1, f2.Choosing e1 to be parallel to X

|X| , we see that βi − αi is an angular function for X|X| with respect

to f1, f2. Hence, we arrive at

2π χ(Σ)=

n

∑i=1

∫∂Vi

(βi − αi

)−(κi

g − κig)

d` =n

∑i=1

∫∂Vi

α(s) d` = 2πn

∑i=1

µ(pi) .

Using the fact that χ(S2) = 2, we now directly obtain Corollary VII.17 from Corollary VII.18:

Proof (of Corollary VII.17) : If X ∈ X(S2) had no zeroes, then there were no stationary pointsand consequently ∑i µ(pi) = 0. But this is a contradiction to Corollary VII.18,

n

∑i=1

µ(pi) = χ(S2) = 2 > 0 ,

hence there is p ∈ S2 so that X(p) = 0.

page 90

A Brief Summary

The following gives a quick overview of what has been covered in each chapter of this course.Closer to the exam, you will also obtain a commented overview, indicating more and less importantparts. (Here, importance is of course measured with respect to the exam, not mathematicalconsequences.)

II. Definitions related to curves, arc length and the Frenet-Serret frame; characterisation ofarc length, the Frenet-Serret equations, geometric meaning of curvature and torsion, theFundamental Theorem of the Local Theory of Curves; simple closed curves, chains and polygonsand the Theorem of Turning Tangents

III. Definitions related to submanifolds; characterisations of submanifolds and its tangentspaces; the differential; special case of surfaces

IV. Definitions of the fundamental forms, the Gauss and Weingarten maps; definitions of thecurvatures (Gauss, mean and principal); computing local expressions for these objects (theroad map); important examples

V. Definitions related to isometries, vector fields and the covariant derivative; characterisationof isometries, examples of (non–)isometric surfaces; properties of the covariant derivative,the Christoffel symbols, example of explicit calculations; the Theorema Egregium and theTheorem of Bonnet

VI. Definitions of parallel vector fields, geodesics and normal and geodesic curvature; proper-ties and examples of geodesics; geometric meaning of normal and geodesic curvature

VII. Definitions of simple closed curves, chains and their angular functions in surfaces; the localTheorems of Gauss-Bonnet; definitions of polygonal covers and the Euler characteristic; theglobal Theorem of Gauss-Bonnet and applications

page 91

karsten fritzsch [email protected] · 2018-10-11 · 2016–17 differential geometry...

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