3 the tangent bundle - arizona state university

Classnotes: Geometry & Control of Dynamical Systems, M. Kawski. February 23, 2009 37

3 The tangent bundle

3.1 Introduction

This is a good time to reflect why we want a notion of tangent spaces and tangent maps in thefirst place. What are such objects supposed to deliver? What properties should tangent vectorsand tangent spaces have? What are the tangent spaces to the line R and the plane R2– two ofthe most familiar manifolds?We want to use our experience with tangent lines to curves and tangent planes to surfaces intwo- and three dimensional Euclidean spaces as guidance. However, in general we do not wantour notion of tangent objects to depend on, or be constrained by imbeddings of the manifoldinto some Euclidean space. Thus without any surrounding space available, the pictorial arrowsbecome untenable. Before reading on, you should close the notes and brainstorm some ideas . . .

Some ideas which come to mind are:

• Tangent objects should implement linear approximations of objects on manifolds and ofmaps between manifolds, where infinitesimally linear is synonymous with differentiable.Recall that we already have notions of derivatives of maps Φ: M 7→ N , but only withrespect to coordinate charts (u,U) and (v, V ), in terms of the maps (vΦu−1) : Rm 7→ Rn.Clearly a coordinate-free notion is desirable.

• Tangent vectors should be vectors, i.e. be members of a linear space that provides foraddition and scalar multiplication.

• The dimension of the linear tangent space(s) should equal the dimension of the manifold.

• Vector fields are intimately connected to differential equations / dynamical systems. Thustangent vectors should provide a means to describe dynamical systems on manifolds, aswell as, more generally, partial differential equations on manifolds. This includes thespecial case of gradient vector fields, which are the derivatives of some potential function.[[This will lead to the cotangent bundle and higher order bundles.]]

• We defined arc-length as an integral of the speed, rather then as the supremum of thelength of polygonal approximations. In general tangent vectors may provide a meanson which to base a generalized notion of distance. [[This is the key idea that leads toRiemannian metrics.]]

• We defined curvature for curves in Rn in terms of the rate of change of the tangent vectors.Thus we expect that a general notion of (comparing) tangent spaces (at different points)should provide for a notion of curvature. [[The notion of curvature will be rephrased interms of comparing tangent spaces at different points.]]

Should our definition allow tangent spaces at different points of a manifold to have nonemptyintersection? E.g. consider the unit-circle S1 imbedded in the plane R2. The tangent lines toS1 at p = (1, 0) and at q = (0, 1) intersect nontrivially at (1, 1). On the other hand, if we thinkof the tangent vectors vp = (0, 1) and vq = (1, 0) as arrows based at p and q respectively, thenwe certainly think of them as different.This brings up a larger issue of distinguishing vectors (arrows) that may be moved around andvectors that are rooted at a fixed point. There are many applications where it is advantageousto consider equivalence classes of directed line-segments, equivalence meaning that they maybe transformed into each other by parallel translation. Compare the first definition as objectscharacterized by their direction and magnitude, but which have no fixed root. On the other


hand, there are many places where it is appropriate to consider vectors that are rooted, or fixedat their base points, velocity vectors to a curve, and more generally vector fields..The next section shall be s first effort to bring clarity to these issues and make very precisedefinitions (which may always be relaxed where this causes no trouble).

3.2 Tangent spaces

There are many different ways in which one may motivate an eventual construction of tangentspaces to a general manifold. One typically starts from surfaces in Euclidean spaces, then consid-ers more abstractly immersed manifolds in higher dimensional Euclidean spaces, and eventuallytries to develop a notion that works in abstract settings, yet reduces to the familiar ones inEuclidean settings. For a lengthy such discussion see Spivak Vol. I ch. 3.An intuitive (and very useful) way to define tangent vectors to a manifold M at a point pis as equivalence classes of curves. Roughly, two curves are equivalent if they have the samevelocity vector at p – but this would be circular as we don’t have notions of velocity vectorsfor general curves on manifolds. So the next best thing is to declare any two smooth curvesσ, γ : (−ε, ε) 7→ M (with σ(0) = γ(0) = p) equivalent if for every smooth function f ∈ C∞(p)(that is, smooth function defined on some neighborhood of p) d

dt |t=0(f σ) = ddt |t=0(f γ).

Project 3.1 Further explore how this leads to a notion of tangent spaces that is basically thesame as the one we define below. In particular, equip the collection of equivalence classes withan addition and scalar multiplication (make sure that these are well-defined). Check that thespace of tangent vectors at a point is indeed an m-dimensional vector space. In a coordinatechart find a basis for the tangent space (e.g. provide representatives (curves) for m equivalenceclasses that form a basis). Show how to write any tangent vector, that is any equivalence classof curves, as a linear combination of this basis. Analyze how the coordinates of a tangent vectortransform under local coordinate changes on the manifold. Match this notion of tangent spacesto the one provided below.

The exercise already hinted at a useful connection between tangent vectors and derivatives.Indeed, going back to Euclidean spaces, say e.g. M = R2 consider the relation between a vector~v = (v1, v2) = v1~ı + v2~ and the directional derivative (operator) D~v|p, defined by D~v|pf =v1 · (D1f)(p) + v2 · (D2f)(p), commonly also written as D~vf(p) = 〈v , ~∇f(p)〉.Clearly, any vector ~v uniquely determines a (directional) derivative operator D~v|p. Conversely,one can easily recover the vector ~v from the directional derivative D~v by simply evaluatingthe latter on suitable functions: For example, evaluating D~v|p on the coordinate functionsπ1, π2 : R2 7→ R defined by π1(x1, x2) = x1 and π2(x1, x2) = x2 immediately recovers thecoordinates of ~v as v1 = (D~vπ

1)(p) and v2 = (D~vπ2)(p). Thus there appears to be no harm in

identifying the vector ~v with the directional derivative operator D~v(·)(p). This idea will provemost beneficial since operators on spaces of functions are automatically endowed with a richalgebraic structure that is ready to use! Following [Chevalley, 1946] define

Definition 3.1 A tangent vector to a manifold M at a point p ∈ M is a functionXp : C∞(p) 7→ R which is linear over R and is a derivation on C∞(p), i.e. which satisfiesfor λ ∈ R and for all f, g ∈ C∞(p) on their common domain,

• Xp(λf + g) = λXp(f) +Xp(g).

• Xp(f · g) = Xp(f) · g(p) + f(p) ·Xp(g)

The set of all tangent vectors to M at p is called the tangent space to M at p, denoted TpM .


This idea of identifying objects such as tangent vectors as operators on thealgebra C∞(M) of smooth functions on M is part of a much larger theme.For example, analogous the above one might consider the multiplicative linearfunctionals p : C∞(M), i.e. which satisfy for all f, g ∈ C∞(M) and c ∈ Rp(f + cg) = p(f) + cp(g) and p(fg) = p(f)p(g). It is easily seen, that eachpoint p ∈ M gives rise to such a functional via p(f) = f(p). More generally,one may start with an algebra A (with suitable topology or norm), and aswhether there exists a manifold M such that A is isomorphic to C∞(M). Acritical observation is that functionals of the above form that correspond toevaluations at a point are associated to maximal ideals – think of the idealof all functions f ∈ C∞(M) which are zero at a fixed point. Indeed, undersuitable technical hypotheses it is indeed possible to recover the manifold fromthe algebra suitably topologizing the space of maximal ideals. But this is justthe beginning of a long story. See e.g. the introduction of the textbookControl Theory from a Geometric Point of View by Agrachev and Sachkov,for a very quick introduction to these tools. Historically, the starting point forthis approach is usually taken to be I. Gelfand’s dissertation in 1935, as wellas related work by Marshall Stone and others that started in the late 1920s.The differential geometry part of this course will only use this point of viewfor tangent vectors, but the control section may make more general use ofsuch technical tools. Important for this course is to frequently reflect on theadvantages and convenience of this operator approach, as well as on its costs.

Exercise (Spivak I 3.13): Use Mp = f ∈ C∞(M) : f(p) = 0 and show that(Mp/M2

p

)∗is a model for TpM (where M2

p = fg : f, g ∈Mp).

Recall, f ∈ C∞(p) means that there exists an open neighborhood U of p (depending on f)such that f ∈ C∞(U,R). (This is a special case of the earlier definition of C∞(A) for subsetsA ⊆ Rn that are not necessarily open.) The following lemma only uses the abstract notion of aderivation, no coordinates are required:

Lemma 3.1 If c : M 7→ R is a constant function and Xp ∈ TpM then (Xpc) = 0

Proof. Use the linearity and Leibniz rule for differentiating products to establish

c ·Xp(1) = Xp(c · 1) = (Xpc) · 1 + c · (Xp1) and hence (Xpc) = 0.

Observations:

• TpM is a vector space: In particular, if Xp, Yp ∈ TpM and λ ∈ R then (Xp + λYp) ∈TpM . The addition and scalar multiplication are inherited from pointwise evaluations, i.e.(Xp + λYp)f = (Xpf) + λ(Ypf).

• Tangent vectors are local operators: If f, g ∈ C∞(M) agree on some neighborhood U of p,i.e. f |U ≡ g|U then for all Xp ∈ TpM , Xpf = Xpg.Technically, the “natural domain” of tangent vectors Xp ∈ TpM are germs of functions atp: The germ of a function f ∈ C∞(p) is defined as the set of all g ∈ C∞(p) for which thereexists an open neighborhood U of p so that f |U ≡ g|U.


• If M = Rm then the tangent vectors to M at a point p are precisely the directionalderivatives evaluated at p. (One direction is obvious. For the other see the calculationbelow for a general manifold.)

• If (u,U) is a chart about p ∈ M , then for j = 1, . . .m, i ∂∂uj

∣∣p∈ TpM . Thus also

m∑j=1

aj ∂∂uj

∣∣p∈ TpM for all aj ∈ R. To verify this assertion, recall that for f ∈ C∞(p)

∂f∂ui

∣∣∣p

= Di(f u−1)∣∣u(p)

and use the familiar properties of partial derivatives in Rn to manipulate e.g.

∂(f ·g)∂ui

∣∣∣p

= Di((f · g) u−1)|u(p)

= Di((f u−1) · (g u−1))|u(p)

= Di(f u−1)|u(p) · (g u−1)(u(p)) + (f u−1)(u(p)) ·Di(g u−1)|u(p)

= ∂f∂ui

∣∣∣p· g(p) + f(p) · ∂g

∂ui

∣∣∣p

More interesting is the converse, i.e. that in any chart (u, U) about p ∈M every tangent vectorXp ∈ TpM may be expressed as a linear combination of the partial derivatives ∂

∂ui , i = 1, . . .m:

Theorem 3.2 If (u, U) is a chart about p ∈Mm then ∂∂u1

∣∣p, . . . ∂

∂um

∣∣p is a basis for TpM .

Corollary 3.3 If (u, U) is a chart about p ∈Mm and Xp ∈ TpM then Xp =m∑

i=1(Xpu

i) · ∂∂ui

∣∣p.

Corollary 3.4 If (u, U) and (v, V ) are charts about p ∈Mm then ∂∂vi

∣∣p

=m∑

i=j

∂uj

∂vi

∣∣∣p· ∂

∂uj

∣∣p.

Consider this last corollary as a statement about (linear) bases changes in the tangent spaceassociated to (nonlinear) changes of local coordinates on the manifold. Before proving thistheorem establish a lemma using an elegant construction

Lemma 3.5 Let (u, U) be a chart about p ∈Mm with u(p) = x0 and f ∈ C∞(p). Then there ex-

ist fi ∈ C∞(p) such that fi(p) = ∂f∂ui

∣∣∣pfor i = 1, . . .m, and f(q) = f(p)+

m∑i=1

(ui(q)− ui(p)

)fi(q).

This is basically a first-order Taylor expansion with remainder term. Here we have equality(as opposed to an approximation) – but the functions fi are evaluated at variable points q asopposed to fixed derivatives evaluated at the fixed point p.Proof (of the lemma). Using the local coordinates we reduce the proof to Euclidean spaces:Write x = u(q) and x0 = u(p), rewrite the second statement of the lemma as

f(u−1(x)) = f(u−1(x0)) +m∑

j=1

(uj(u−1(x))− uj(u−1(x0))

)fj(u−1(x))


Write g for f u−1 : u(U) ⊆ Rm 7→ R. The desired functions gi : u(U) ⊆ Rm 7→ R are such that

g(x) = g(x0) +m∑

j=1

(xj − xj0)gj(x).

After shrinking the neighborhood U , if necessary, we may assume that u(U) ⊆ Rm is star-shaped with respect to x0, i.e. for every x ∈ u(U) the line segment x0 + t · (x− x0) : t ∈ [0, 1]is contained in u(U). For any fixed x ∈ u(U) consider the curve σx : [0, 1] 7→ u(U) defined byσ(t) = x0 + t · (x− x0). Via the fundamental theorem of calculus and the chain rule

g(x) = g(σx(1)) = g(σx(0)) +∫ 10

ddtg(σx(t)) dt

= g(σx(0)) +1∫0

m∑j=1

(Djg)(σx(t)) · dσjx

dt (t) dt

= g(σx(0)) +m∑

j=1(xj − xj

0)∫ 1

0(Djg)(σx(t)) dt︸︷︷︸

define as gj(x)

Note the constant derivative σ′x = (x− x0) for the curve in Rm – this makes no sense on a gen-eral manifold, but is coordinate dependent. One immediately verifies that with this definitiongj(x0) = (Djg)(x0) (in this case σx0 is a constant curve). Consequently fj(p) = gj(u(p)) =

(Djg)(u(p)) = Dj(f u−1)|u(p)= ∂f

∂uj

∣∣∣p. Since g ∈ C∞, also gj ∈ C∞ and fj ∈ C∞.

Proof (of the theorem). Suppose (u, U) is a chart about p ∈Mm, Xp ∈ TpM and f ∈ C∞(M).Using lemma 3.5 there exist (on some open neighborhood of p) suitable functions fi such that

we may rewrite Xpf as Xpf = Xp

(f(p) +

m∑j=1

(uj − uj(p)

)fj

). Using the linearity of Xp, the

Leibniz rule, and that Xpc = 0 for any constant function this yields:

Xpf = 0 +m∑

j=1

(Xp(uj)− 0)fj(p) + (uj(p)− uj(p))︸︷︷︸=0

·(Xpf)

,

i.e. (Xpf) =∑m

j=1(Xpui) · fi(p). By lemma 3.5 this is equal to (Xpf) =

∑mj=1(Xpu

i) ∂f∂ui

∣∣∣p.

Since this holds for all f ∈ C∞(p), we conclude Xp =∑m

j=1(Xpui) ∂

∂ui

∣∣p.

3.3 Tangent maps, part I

For every smooth map F : Rn 7→ Rm between Euclidean spaces and any point x ∈ Rn thederivative of F at x is a linear map (DF )(x)(·) : Rn 7→ Rm. Now that we have tangent spaces tomanifolds, we are ready to associate analogous (linear) tangent maps (between tangent spaces)to smooth maps (between manifolds).

Definition 3.2 Suppose Φ: M 7→ N is a smooth map between manifolds and p ∈ M . Thetangent map Φ∗p : TpM 7→ TΦ(p)N (of Φ at p) is defined for Xp ∈ TpM and f ∈ C∞(Φ(p)) by

(Φ∗pXp) f = Xp(f Φ). (39)


How else could Φ∗p be defined? It is immediate that if Φ = idM then Φ∗p = idTpM . Also, itfollows immediately from the definition that

Proposition 3.6 Suppose Φ: Mm 7→ Nn and Ψ: N 7→ P are smooth maps between manifolds,and p ∈M . Then (note the preservation of the order of Φ and Ψ)

(Ψ Φ)∗p = Ψ∗Φ(p) Φ∗p. (40)

Exercise 3.2 Prove proposition 3.6.

In local coordinates the tangent map is given by matrix multiplication. More specifically, suppose(u, U) and (v, V ) are charts about p ∈ Mm and Φ(p) ∈ Nn, respectively, and f ∈ C∞(Φ(p)).From the definitions calculate (using the chain-rule)(

Φ∗p ∂∂uj

∣∣p

)f = ∂

∂uj

∣∣p(f Φ)

= Dj(f Φ u−1)|u(p)

= Dj((f v−1) (v Φ u−1))|u(p)

=n∑

i=1Di(f v−1)|v(p) ·Dj(vi Φ u−1)|u(p)

=(

n∑i=1

∂(viΦ)∂uj

∣∣∣p· ∂

∂vi

∣∣Φ(p)

)f

In concrete examples, using local coordinates, it is convenient to express tangent vectors ascolumn vectors. E.g. suppose, as before, Φ ∈ C∞(Mm, Nn), and (u, U) and (v, V ) are chartsabout p ∈ Mm and Φ(p) ∈ Nn. If Xp ∈ TpM is a tangent vector at p, let a = (a1, . . . am)T bethe column vector with components ai = Xpu

i, representing Xp =∑m

i=1 ai ∂

∂ui

∣∣p. Similarly, let

b = (b1, . . . bn)T be the column vector with components bj = (Φ∗pXp)vj , representing the imageΦ∗pXp ∈ TΦ(p)N . These column vectors a and b are related by matrix multiplication b = Ca

where C is the (n×m) matrix with components Cij = ∂(viΦ)∂uj

∣∣∣p.

Formally, one may go further, and write α for the row-vector with components αi = Φ∗p ∂∂ui

∣∣p

and β for the row-vector with components βj = ∂∂vj

∣∣Φ(p)

. Then formally, the images of the basis

vectors ∂∂ui |p are obtained by right matrix multiplication, i.e. α = β · C. This matches with

the observation that formally Φ∗p(Xp) = αa = (βC)a = β(Ca) = βb. While this is all simple(formal) matrix algebra, it is worthwhile to remember that when transforming formal vectorsof basis elements these are multiplied by the transformation matrix in a way opposite to themultiplication familiar for transforming specific vectors.

Exercise 3.3 Suppose Φ ∈ C∞(Mm, Nn) and Ψ ∈ C∞(Nn, P r) are smooth maps betweenmanifolds, p ∈M and f ∈ C∞(P ).Furthermore, suppose (u,U), (v, V ) and (w,W ) are local coordinate charts about p ∈M , Φ(p) ∈N and (Ψ Φ)(p) ∈ P , respectively. Verify that the matrix representing (Ψ Φ)∗p with respectto (u, U) and (w,W ) is the (matrix-)product of the matrices representing Φ∗p (with respect to(u, U) and (v, V )) and Ψ∗Φ(p) (with respect to (v, V ) and (w,W )).

We digress a little to consider tangent spaces of immersed manifolds in Rn which justify thefamiliar pictures of tangent planes. Suppose that Φ ∈ C∞(Mm,Rn) is an immersion at p ∈M ,


i.e. rankpΦ = m. Using local coordinates (u, U) about p ∈ Mm and the standard coordinates(x,Rn) in the codomain, the rank condition says that the (n × m)-matrix with components∂(xiΦ)

∂uj

∣∣∣p

has rank m, and Φ∗p is a monomorphism (a linear one-to-one map) from TpM to

TΦ(p)Rn.

The tangent vectors Φ∗p(

∂∂uj

∣∣p

)∈ Φ∗p (TpM) ⊆ TΦ(p)Rn are linearly independent, and span an

m-dimensional subspace of TΦ(p)Rn which is usually pictured as a tangent line/plane.The image of any tangent vector Xp ∈ TpM in the standard coordinates may be written in theform Φ∗pXp =

∑ni=1 b

iDi|Φ(p). Now, if f ∈ C∞(Rn) is a function such that f Φ ≡ 0 in aneighborhood of p ∈M , then

0 ≡ Xp(f Φ) = (Φ∗pXp)f =n∑

i=1

bi (Dif)|Φ(p) (41)

which in calculus notation might be written as Φ∗pXp ⊥ (grad f)(Φ(p)), or 0 = 〈b , (~∇f)Φ(p))〉.Recall if (gradf)(Φ(p)) 6= 0, then f−1(0) is (locally) a smooth hypersurface in Rn(near Φ(p).Thus (Φ∗pXp) may be pictured as lying in the tangent hyperplane to the hypersurface f−1(0)at Φ(p).

Exercise 3.4 Generalize this discussion to the case when there are functions f1, . . . fn−m ∈C∞(Φ(p)) with f i · Φ ≡ 0 and linearly independent gradients (gradf i)(Φ(p)).

Example 3.1 As a hands-on example consider an immersion of the M-strip into R3. One wayto represent the M-strip is as the quotient M = R/ ∼ of the rectangle R = [0, 2π] × (−1, 1)two of whose edges have been identified by (0, t) ∼ (2π,−t). Define a map Φ: M 7→ R3 byΦ(θ, t) =

((2 + t cos(1

2θ)) cos θ , (2 + t cos(12θ)) sin θ , t sin(1

2θ)).

Exercise 3.5 Explicitly calculate the images Φ∗p ∂∂θ

∣∣p

and Φ∗p ∂∂t

∣∣p

at any pointp = (θ, t) ∈ U = (0, 2π)× (−1, 1) ⊆M . Verify that Φ is indeed an immersion.

The image Φ(M) ⊆ R3 is a surface in the usual sense. The images of the tangent vectors to Mcalculated in the exercise may be visualized as the usual arrows that are tangent to the surface.It is easily seen that the map Φ is indeed well-defined on M (as opposed to only on the rectangleR) because Φ(0, t) = Φ(2π,−t). For p ∈ U the map Φ∗p is well-defined, but problems arise whentrying to extend Φ and Φ∗q continuously to all q ∈ M . Indeed, ((θ, t), U) is a local coordinatechart of M , but it does not cover all of M . In the language of the next sections Φ∗ maps thecoordinate vector fields ∂

∂u1 = ∂∂θ and ∂

∂u2 = ∂∂t (which are only defined on U) to vector fields on

Φ(U)), but the vector field ∂∂u1 cannot be continuously extended to a vector field on all of M .

We conclude this first section on tangent maps with a generalization of the earlier local submer-sion theorem.

Theorem 3.7 Suppose that Φ ∈ C∞(Mm, Nn) is a smooth map between manifolds and P r ⊆ Nis a smooth submanifold. If for every p ∈ Φ−1(P )

Φ∗p(TpM) + TΦ(p)P = TΦ(p)N (42)

then Φ−1(P ) ⊆M is a submanifold of M of dimension (m− (n− r)).


In general one calls a smooth map Φ ∈ C∞(M,N) between manifolds transverse to (a submani-fold) P ⊆ N along (a submanifold) L ⊆M if Φ∗p(TpM)+TΦ(p)P = TΦ(p)N for all p ∈ L∩Φ−1(P ).

The theorem motivates the notion of codimensions as opposed to dimensions of submanifolds.More specifically, for an r-dimensional submanifold P r ⊆ Nn of an n-dimensional manifold Nthe codimension of P (in N) is defined as (n−r). The theorem then simply states that if P ⊆ Nis a submanifold of codimension k and Φ is transversal to P (along M) then Φ−1(P ) ⊆ M is asubmanifold of the same codimension k.

Proof. Suppose that Φ ∈ C∞(Mm, Nn) and p ∈ Φ−1(P ) ⊆ M is in the preimage of asmooth submanifold P r ⊆ N . Using theorem 2.13 choose an adapted chart (v, V ) aboutΦ(p) ∈ N such that v(Φ(p)) = 0 and such that the restriction of w = (v1, . . . vr) to the setW = q ∈ N : vr+1(q) = . . . = vn(q) = 0 is a chart of P about Φ(p).Define Ψ: V 7→ Rn−p by Ψ = (vr+1, . . . vn). Let U = Φ−1(V ) ⊆M .Then p ∈ Φ−1 Ψ−1(0) = (Ψ Φ)−1(0) and (Ψ Φ)∗p = Ψ∗Φ(p) Φ∗p : TpM 7→ T0Rn−r.Use that D(Ψ v−1) = ( 0 In−r ) and that the kernel of Ψ∗Φ(p) is precisely TΦ(p). Togetherwith Φ∗p(TpM)+TΦ(p)P = TΦ(p)N this establishes that the restriction of Ψ∗Φ(p) to the image ofΦ∗p (i.e. to Φ∗p(TpM)) has full rank, and hence rank(ψ Φ)∗p = n− p (using corollary 2.14).

Exercise 3.6 Revisit the Hopf map Φ: S3 7→ S2 (of exercise 2.31), i.e., the restriction (to S3)of Φ : R4 7→ R3 defined by Φ(a) = (2a1a3 + 2a2a4, 2a2a3 − 2a1a4, a

21 + a2

2 − a23 − a2

4).Considering the usual imbeddings of the spheres into R4 and R3, respectively, describe the preim-ages Φ−1(Pc) of the meridians Pc = S2 ∩ (x1, x2, x3) ∈ R3 : x3 = c for −1 ≤ c ≤ 1.


3.4 The tangent bundle

It is natural to assemble all tangent spaces of a manifold together into a new object – andconceivably, this set should again have a natural manifold structure. We will omit some of themore technical details of this construction and refer to e.g. Spivak vol.I ch.3, especially exercise1. As discussed earlier, it is desirable to distinguish between tangent vectors at different points,and we define:

Definition 3.3 The tangent bundle TM of a manifold M is (as a set) the (disjoint) union ofall tangent spaces to M at all points p ∈M .

TM =

(p,Xp) ∈M ×⋃

p∈M

TpM : Xp ∈ TpM

. (43)

The bundle projection π : TM 7→ M is defined by π(p,Xp) = p. The fiber over p ∈ M is thepreimage π−1(p) = p × TpM .A section of TM , or tangent vector field, is a map X : M 7→ TM that satisfies π X = idM .

Basically, vector fields are functions that assign to each p ∈M a tangent vector in TpM .Often we conveniently identify the fiber π−1(p) with TpM and the pair (p,Xp) ∈ TM withtangent vector X(p) = Xp ∈ TpM . Technically this involves a tacit projection onto the secondfactor or a tacit use of the inclusion map ıp : TpM 7→ p × TpM ⊆ TM . However, in someinstances, e.g. when working with TTM , more precision is indicated.

To illustrate that tangent bundles of manifolds are candidates to be considered manifolds them-selves, consider the example of M = S1. The naive collection of all tangent lines to the imbeddedcircle S1 ⊆ R2 is full of intersections. More suitable for our purposes is to imbed the circle inR3 as S1 = x ∈ R3 : x2

1 + x22 = 1, x3 = 0 and attach at every point p ∈ S1 a vertical (!)

line, yielding a cylinder. As a set, this cylinder is in bijection with the (disjoint) collection ofall tangent lines to the circle imbedded in the plane. Intuitively one can consistently choosean orientation of the lines, and even more a consistent scaling. For example, identify the naivetangent vector

((cos Θ, sinΘ), (−L sinΘ, L cos Θ)

)with the point (cos Θ, sinΘ, L) ∈ R3.

Within this picture, a vector field on the circle may be visualized as the graph of a function(cos Θ, sinΘ) 7→ L(θ). If the vector field is continuous and nonvanishing, then the graph liesentirely above, or entirely below the plane x3 = 0.

In analogy, we may intuitively think of the tangent bundle TR of the real line R as the plane R2.However, due to dimensional reasons it is clear that these two examples are the only tangentbundles amenable to such immediate visualization. How quickly things get complicated becomesclear if one tries to think of TS2 as a sphere with a plane attached to each of its points. A vectorfield on the sphere simply selects one point on each plane. However, from algebraic topology itis known that there does not exist any continuous (yet to be defined !) vector field on the spherethat vanishes nowhere. In our picture this means that it is impossible to continuously selectone point on each tangent plane avoiding the origin (zero-vector) in each TpS

2. Intuitively TS2

must be a nontrivially twisted, (when compared to e.g. TS1 which is the very tame cylinder),i.e. it must be very different from the trivial Cartesian product S2 × R2.


We proceed more abstractly to endow the tangent bundle TM of a smooth manifold M witha manifold structure. The key idea is that locally, above a chart (u, U) (which itself is homeo-morphic to Rm) the tangent bundle basically looks like Rm × Rm ∼= R2m. This observation iscaptured in the concept of local triviality, i.e., if (u, U) is a chart of an m-dimensional manifoldM , then TU ∼= U × Rm. Thus the topology and geometry of M are captured in the globalstructure of the tangent bundle, by how the trivial bundles are pieced together in some twistedway.Starting with an atlas of local coordinate charts for M we shall find it easy to obtain a candi-date atlas of charts for TM . There is a natural candidate for a topology on TM that makesall candidate charts for TM into homeomorphisms, and automatically guarantees that theirtransition maps are smooth. However, it takes a little more advanced arguments to show thatthis topology is indeed sufficiently nice (metrizable or paracompact) so that TM qualifies as amanifold.

Suppose (u, U) is a chart of M about p. Consider the subset U = π−1(U) ⊆ TM . Every pointQ ∈ U is a pair Q = (q,Xq) with Xq ∈ TqM . Since ∂

∂uj |q : 1 ≤ j ≤ m is a basis for TqM thereexists functions wj : U 7→ R (indeed, wj(Q) = Xqu

j) such that

Q =(π(Q),

m∑j=1

wj(Q) ∂∂uj

∣∣π(Q)

). (44)

It is natural to define a map u : U 7→ R2m by

u =(u1 π, . . . um π,w1, . . . wm

). (45)

It is clear that u is a bijection from U to R2m. (This assumes that u is a bijection from U toRm, as originally mandated. Alternatively, u is a bijection from U onto u(U)× Rm ⊆ R2m.)

For each chart (u, U) in the C∞-differentiable structure of M , we want the associated map uto be a homeomorphism. This is achieved by endowing TM with the weakest (i.e., coarsest)topology in which all maps u are continuous. More constructively, we consider the collection Tof all subsets O ⊆ TM which are such that for every point Q ∈ TM and every chart (u, U) of Mfor which π(Q) ∈ U , there exists an open set W ∈ R2m containing u(Q) such that u−1(W ) ⊆ O.

Exercise 3.7 Show that the collection T of subsets of TM is a topology on TM , i.e., show that∅, TM ∈ T , and that T is closed under finite intersections and arbitrary unions.

Exercise 3.8 Check that when TM is endowed with this topology T then for every chart (u, U)the associated map u is a homeomorphism, i.e. is continuous with continuous inverse.

Next calculate the transition maps (v u−1) : u(U ∩ V ) ⊆ R2m 7→ v(U ∩ V ) ⊆ R2m to verify thatthe charts are indeed C∞-related.Thus assume that Q ∈ U ∩ V ⊆ TM and that u(Q) = (x, y) ∈ Rm × Rm. Calculate

v(Q) = (v u−1)(x, y) = v(u−1(x),m∑

j=1

yj ∂∂uj

∣∣u−1(x)

). (46)


To obtain the second m-components of v(Q) change the basis in Tπ(Q)M from the

∂∂uj |π(Q)

m

j=1

to the

∂∂vi |π(Q)

m

j=1, yielding

(v u−1)(x, y) = v

(u−1(x),m∑

j=1

yjm∑

i=1

∂vi

∂uj

∣∣∣∣u−1(x)

∂

∂vi

∣∣∣∣u−1(x)

. (47)

Interchange the order of summation and regroup to read off the components of v(Q)

(v u−1)(x, y) = ((v u−1)(x),m∑

j=1

yj ∂v1

∂uj

∣∣∣∣u−1(x)

, . . . ,m∑

j=1

yj ∂vm

∂uj

∣∣∣∣u−1(x)

). (48)

It is easily seen that the map (v u−1) : u(U ∩ V ) ⊆ R2m 7→ v(U ∩ V ) ⊆ R2m is a smooth map:By hypothesis the first m-components are smooth maps. The second m components are linearin y and smooth functions of x, and hence the combined map is smooth. If working in the classof Cr-manifolds, this calculation shows that when one starts with a Cr atlas for M , then oneobtains, as might be expected, a Cr−1 atlas for TM .

In order for TM to qualify as a smooth manifold we still need that the topology is reasonablynice (metrizable, or equivalently that (TM, T ) is paracompact). For the technical details werefer to Spivak vol.I ch.3, especially exercise 1. Here we sketch only some basic ideas. Since amanifold M is locally homeomorphic to a Euclidean space and it is assumed to be metrizable (orequivalently paracompact), each connected component of M is second countable. [[This meansthat there is a countable basis for the topology on each connected component of M . A basisfor a topology is a collection of open sets that covers the space, and such that whenever a pointx is contained in basic open sets B1 and B2, then there exists a basic open set B3 such thatx ∈ B3 ⊆ B1 ∩ B2.]] Following Spivak vol.I ch.3 ex. 1, construct a sequence of functions thatseparates points and closed sets, use these to produce a sequence of bounded metrics di andfinally piece these together e.g. via d =

∑∞i=1 2−idi.

This construction of the tangent bundle shall serve as a model for similar constructions of moregeneral vector bundles in which the tangent spaces TpM are replaced by other suitable vectorspaces. Formally, a vector bundle is a triple (E,B, π) (or actually, a five-tuple (E,B, π,⊕,))consisting of a total space E, a base space B and a bundle projection π : E 7→ B which is acontinuous surjective map. The linear operations ⊕ and are defined on the fibres π−1(p) forp ∈ B, making each fibre a vector space. A distinguishing condition is that a vector bundleis locally trivial, i.e. every p ∈ B has an open neighborhood U together with a homeomor-phism β : π−1(U) 7→ U × Rm such that for each q ∈ U the restriction β|π−1(q)

is a vector space

isomorphism from the fibre π−1(q) to q × Rm.The Mobius strip is an example of a nontrivial line-bundle over the circle S1. An upcomingsection will introduce the cotangent bundle in which the fibre are the spaces of linear functionalson the corresponding tangent spaces. In some places it is convenient to work with functionsthat assign to each point p ∈ M a pair, triple, or m-tuple of (co)-tangent vectors. These maybe thought off as sections of bundles in which each fibre is a product of two, three, or n copiesof the (co)tangent space.Beyond vector bundles are fibre bundles in which the fibres need not necessarily be vectorspaces. Arguably one of the most important ones is the principal bundle in which each fibre isa copy of the general linear group GL(m,R) (the space of all invertible linear maps from Rm to


Rm). Its distinguishing feature is that each section L : M 7→ P (with π L = idM ) acts on themanifold, e. g., if (u, U) is a local coordinate chart, then (L u,U) is another chart (to be readas (L u : q 7→ Lq u(q), U) where Lq : Rm 7→ Rm is a linear map.)

To be added: Use tangent bundles for a geometric definition of orientability for a manifoldM , or of vector bundle - as opposed to the purely algebraic condition in terms of charts,whether there exists atlas A such that det(D(v u−1) > 0 for all (u, U), (v, V ) ∈ A.

We digress with a brief discussion of the (lack of) triviality of the tangent bundles of spheresand its consequences. Consider the usual imbeddings of the spheres Sm → Rm+1 and use thestandard coordinates in Rm+1. Note that the tangent bundles TSm are diffeomorphic to thesubsets (a, b) ∈ Sm × Rm+1 : 〈a , b〉 = 0 ⊆ Rm+1 × Rm+1, using the standard inner productin Rm+1. [[This is completely different from asserting that TSm were trivial, or diffeomorphicto Sm × Rm.]]When m = (2k−1) is odd, then X = x2D1−x1D2 +x4D3−x3D4 + . . . x2kD2k−1−x2k−1D2k (allDj evaluated at x) is a global nonvanishing (tangent) vector field on Sm ⊆ Rm+1. Conversely,using tools from algebraic topology one may show that if m = 2k is even, then not even a singleglobally defined continuous nonvanishing vector field exists on the sphere S2k. Very special isS3 which admits three smooth vector fields that are everywhere linearly independent:

X = −x2D1 + x1D2 − x4D3 + x3D4

Y = −x3D1 + x4D2 + x1D3 − x2D4

Z = −x4D1 − x3D2 + x2D3 + x1D4.(49)

Mimicking (and repeating this construction, similar to the example for S2k−1 above) one mayconstruct from these three vector fields on S3 three everywhere linearly independent vector fieldson any sphere S4k−1. However, it can be shown that on S4k+1 any two smooth vector fields arelinearly dependent at some point. This example of the frame of the three everywhere linearlyindependent vector fields on S3 motivates the notion of a parallelizable manifolds [[Abraham-Marsden p.218; Boothby p.219; not in Spivak]]:

Definition 3.4 A manifold Mm is called parallelizable if it admits a frame of m everywherelinearly independent vector fields.

It is straightforward to see that a [[finite-dimensional, c.f. Abraham-Marsden]] manifold isparallelizable if and only if its tangent bundle is trivial. So far we have seen that Euclidean spacesRm, hence all coordinate charts (u, U) are parallelizable. Also, every Lie group is parallelizable.

Should have been done much earlier, in chapter 2: A Lie group is a differentiablemanifold G with a group structure such that both the multiplication : G×G 7→ G definedby (p, q) 7→ pq and the inverse G 7→ G, defined by p 7→ p−1 are C∞ maps.We already encountered several examples of Lie groups: the general linear groups GL(n,R)of invertible linear maps on Rn, the special linear groups SL(n,R) (linear maps withdeterminant one), and the orthogonal groups O(n) and special orthogonal groups SO(n).Note that as a manifold S1 is diffeomorphic to SO(2). Similarly, S3 is a double-cover ofthe projective space P 3 which is diffeomorphic to SO(3) – thus shedding some light onthis most versatile example. In quantum mechanics these Lie groups are supplanted bytheir complex sisters, the unitary U(n) and special unitary groups SU(n).


Exercise 3.9 Suppose f ∈ C∞(Rm,R). Show that the graph (x, f(x)) : x ∈ Rm is a paral-lelizable submanifold of Rm+1. Is the same necessarily true for functions f : Rm 7→ Rn?

Returning to the tangent bundles of the spheres: It is known that the only parallelizable spheresare S1, S3, and S7 [[Spivak vol.I, ch.3, ex. 19]]. It is no coincidence that these are the onlydimensions in which one may endow the Euclidean spaces with some sort of a multiplicativestructure: In R2 this is the field of complex numbers, in R4 this yields the noncommutative, butstill associative quaternions (or Hamilton numbers), and in R8 these are the Cayley numberswhose multiplication is not even associative.

3.5 Smooth vector fields and Lie products

Recall, a vector field on a manifold is defined as a section of the tangent bundle, that is, a functionX : M 7→ TM such that its composition π X with the bundle projection is the identity on M .Rather than considering arbitrary such functions, our interest is primarily in those that varysmoothly (in topological considerations continuity may suffice). Since a vector field is definedas a map between manifolds M and TM we already have a notion of smoothness: A vector fieldX : M 7→ TM is Cr if for every point p ∈ M and coordinate charts (u, U) and (v, V ) about pand X(p), respectively, the map v X u−1 : Rm 7→ R2m is a Cr-map between Euclidean spaces.We write Γ∞(M) for the set of all smooth vector fields on M .On the other hand recall that every tangent vector Xp ∈ TpM maps C∞(p) 7→ R. Consequently,we may view a vector field X as a mapping of the algebra C∞(M) of smooth functions toitself. We expect that if X is a smooth vector field and f ∈ C∞(M) then (Xf) ∈ C∞(M).In particular, any smooth vector field is a derivation on the algebra C∞(M) (i.e. it satisfiesX(fg) = (Xf)g + f(Xg) for all f, g ∈ C∞(M)).

Proposition 3.8 A vector field X : M 7→ TM is a C∞ vector field, written X ∈ Γ∞(M), if andonly if for every open set U ⊆M and every function f ∈ C∞(U) the function (Xf) : p 7→ X(p)fis again in C∞(U).

This proposition follows easily from the following exercise upon expanding (Xf)(p) on a chart(u, U) about p in terms of local coordinates (Xf)(q) =

∑mj=1(Xu

j)(q) ∂f∂uj

∣∣∣q

and writing out the

components of the map u X u−1 : Rm 7→ R2m.

Exercise 3.10 Verify directly that a vector field X : M 7→ TM is C∞ if and only if for everycoordinate chart (u, U) of M the functions Xuj : U 7→ R are smooth.

Since every (smooth) vector field X ∈ Γ∞(M) maps C∞(M) back into itself, it is natural toconsider compositions of two vector fields X,Y ∈ Γ∞(M). Clearly X Y , also written XY , isagain a map from C∞(M) into itself. However, for two functions f, g ∈ C∞(M) we calculate

XY (fg) = X((Yf)g + f(Yg)

)= (XYf)g + (Yf)(Xg) + (Xf)(Yg) + f(XYg). (50)

In general there is no reason for the terms (Yf)(Xg) and (Xf)(Yg) to cancel each other, hencein general XY is not a derivation, and thus is not a vector field! However, the commutatorXY − Y X clearly will be a derivation. Thus we have a product structure on Γ∞(M) – whichequips this space of all smooth vector field with an important algebraic structure that invitesdeeper study:


Definition 3.5 The Lie bracket or Lie product of vector fields is the map[ · , · ] : Γ∞(M)× Γ∞(M) 7→ Γ∞(M), defined for f ∈ C∞(M) by [X,Y ]f = X(Yf)− Y (Xf).

Exercise 3.11 Let ξ = (ξ1, . . . ξm)T and η = (η1, . . . ηm)T be column vector fields representingtwo vector fields X,Y ∈ Γ∞(M) in a coordinate chart (u, U), i.e. ξj = (Xuj) and ηj = (Y uj).Verify that in these coordinates the Lie product [X,Y ] is represented by the column vector (Dη)ξ−(Dξ)η where D denotes the Jacobian matrix of partial derivatives.

Definition 3.6 A linear vector space L equipped with a bilinear mapping [ · , · ] : L× L 7→ Lis a Lie algebra if this map is anti-commutative and satisfies the Jacobi identity:

for all x, y ∈ L, 0 = [x, y] + [y, x]for all x, y, z ∈ L, 0 = [x, [y, z]] + [y, [z, x]] + [z, [x, y]]

. (51)

Exercise 3.12 Verify that R3equipped with the standard cross-product is a Lie algebra.

Exercise 3.13 Verify that the space so(3) of skew symmetric 3 × 3-matrices with the product[A,B] = AB −BA (matrix product) is a three dimensional Lie algebra.Find a basis for so(3) and establish a Lie algebra isomorphism from so(3) to R3with the cross-product – i.e. explicitly give a bijective linear map (between vector spaces) that is also a Liealgebra homomorphism, meaning in this case Φ([A,B]) = Φ(A)× Φ(B) for all A,B ∈ so(3).

Exercise 3.14 Verify by direct calculation that the Lie product of vector fields as defined aboveequips Γ∞(M) with a Lie algebra structure. Note that this means verifying that [ · , · ] is linearover R, i.e. [aX + Y, Z = a[X,Z] + [Y, Z], that it is anticommutative (obvious) and that itsatisfies the Jacobi identity – simply expand [X, [Y, Z]]f + [Y, [Z,X]]f + [Z, [X,Y ]]f .

Exercise 3.15 Show that any associative algebra (A, ·) is a Lie algebra with the commutatorproduct that is defined by [x, y] = x · y − y · x. In particular, the set D(A) of derivations onan associative algebra, that is of linear maps ` : A 7→ A satisfying `(xy) = (`(x))y + x`(y)for all x, y ∈ A is an associative algebra under composition and thus a Lie algebra under thecommutator as above.

Exercise 3.16 (This is a preview of an example which geometrically is situated in the cotangentbundle and symplectic geometry). On the set of all smooth functions C∞(R2m) define a product,the Poisson bracket, using coordinates (q1, . . . , qm, p1, . . . pm) on R2m by

for all f, g ∈ C∞(R2m), f, g =m∑

i=1

∂f

∂pi

∂g

∂qi− ∂f

∂qi

∂g

∂pi. (52)

Verify that this product equips C∞(R2m) with the structure of a Lie algebra.

Recall that if X,Y ∈ Γ∞(M) and f, g ∈ C∞(M) then fX ∈ Γ∞(M), and the usual distributiveand mixed associative properties hold, e.g. (fg)X = f(gX), f(X +Y ) = fX + fY , (f + g)X =fX + gX, 1 · X = X. This means that Γ∞(M) is not only a vector space over R, but also a(left) C∞(M)-module. (It is not a vector space over C∞(M) since the ring of smooth functions


is not a field.) Given this C∞(M)-module structure it is natural to ask how the Lie bracket onΓ∞(M) relates to it. For X,Y ∈ Γ∞(M) and f, g ∈ C∞(M) calculate

[fX, Y ]g = (fX)(Yg)− Y (fX(g))

= f(X(Yg)− Y (Xg)

)− (Yf) · (Xg)

=(f [X,Y ]− (Yf)X

)g

(53)

and conclude that the Lie bracket [ · , · ] is not linear over C∞(M).In a chart (u, U) (compare also exercise 3.20) calculate[

∂∂ui ,

∂∂uj

]f = ∂

∂ui (Dj(f u−1) u)− ∂∂uj (Di(f u−1) u)

= Di(Dj(f u−1) u u−1) u− ∂∂uj (Di(f u−1) u u−1) u

=(DiDj(f u−1)−DjDi(f u−1)

) u

≡ 0

(54)

since the mixed partial derivatives on C∞Rm are equal. As an important corollary we obtain:

Proposition 3.9 If X,Y ∈ Γ∞(M) and U ⊆M is an open set with [X,Y ]|U 6≡ 0 then there doesnot exist a map u : U 7→ Rm such that (u, U) is a chart on M with X|U = ∂

∂u1 and Y |U = ∂∂u2 .

Indeed, in subsequent sections we will see that in the neighborhood of any point p at which asmooth vector field X does not vanish, there are always coordinates (u, U) such that X = ∂

∂u1 .On the other hand, generalizing the above criterion to sets of vector fields will lead to importantFrobenius integrability theorem.

Exercise 3.17 [[This exercise is somewhat frivolous – but it is a good practice for hands-on calculations, and it hits hard at common misperceptions.]] Consider the upper half planeM = (x1, x2) ∈ R2 : x2 > 0 with standard rectangular coordinates (x, y), with polar coordinates(r, θ) and with the mixed coordinates (ρ, ξ) defined by ρ = r and ξ = x.

• Explicitly express the coordinate vector fields ∂∂r and ∂

∂θ as linear combinations of ∂∂x and

∂∂y . (In particular express the coefficients in terms of x and y).

• Use these expressions to verify by direct calculation that [ ∂∂r ,

∂∂θ ] ≡ 0.

• Verify by direct calculation that [ ∂∂r ,

∂∂x ] 6≡ 0.

• Explain why this does not contradict that (ξ, ρ) = (x, r) are admissible local coordinates.Calculate the (ξ, ρ) coordinates of the points (1, 0.1), (1, 1), and (0, 1).

• Calculate ∂∂ξ and ∂

∂ρ , e.g. write these as linear combinations of ∂∂x and ∂

∂y , and sketch thesecoordinate vector fields as arrows in the half-plane. Describe in words in which directionsthese arrows point.

• Explain where such possible misconceptions might come from. In some sense, for partialderivatives it is less important what varies than what held fixed . . .Revisit this discussionlater using differential forms dx, dξ, dρ, dη.


3.6 The tangent map and vector fields

Having assembled the tangent spaces TpM at all points p ∈M into the tangent bundle TM asa manifold, it is natural to combine the tangent maps Φ∗p associated to a map Φ ∈ C∞(M,N)between manifolds into a map Φ∗ : TM 7→ TN . This is a straightforward definition with no orfew ensuing surprises. However, in general, tangent maps need not map vector fields to vectorfields.

Definition 3.7 For any map Φ ∈ C∞(M,N) define an associated tangent map Φ∗ : TM 7→ TNfor q ∈M, (q,Xq) ∈ π−1(q) by

Φ∗(q,Xq) = (Φ(q),Φ∗q(Xq)). (55)

Note that the tangent map Φ∗ has the map Φ built in. It is straightforward to verify thefollowing:

Proposition 3.10 If Φ ∈ C∞(M,N) and Ψ ∈ C∞(N,P ) then (note preservation of order)

(Ψ Φ)∗ = Ψ∗ Φ∗. (56)

Exercise 3.18 Show that if Φ ∈ C∞(M,N) then Φ∗ ∈ C∞(TM,TN). (Use the definition ofdifferentiability of a map between manifolds in terms of charts (u, U) and (v, V ) for M and N ,respectively.)

It is important to understand that in general there is no hope that a tangent map associatedto a smooth Φ: M 7→ N between manifold will map a vector field X on M to a vector field onN . This is immediately clear if we recall that a vector field on N is a function from N to TN .Thus if p1 6= p2 ∈ M but Φ(p1) = Φ(p2) ∈ N then problems arise unless Φ∗p1Xp1 = Φ∗p2Xp2 .Similarly, if Φ is not onto, then Φ∗ can at best yield a partially defined vector field on N .If Φ: M 7→ N is a diffeomorphism and X ∈ Γ∞(M) then, with some abuse of notation, defineΦ∗X : N 7→ TN by

(Φ∗X)q = Φ∗Φ−1(q)XΦ−1(q) for q ∈ N. (57)

Sometimes this is written suggestively as (Φ∗X)q = (Φ∗XΦ−1)(q). It is clear that π(Φ∗X) =idN and that Φ∗ is a smooth map, and hence (Φ∗X) ∈ Γ∞(N).An important application is when Φ = u : U 7→ Rm is a coordinate map. Indeed, we haveroutinely used the map u∗ which maps e.g. coordinate vector fields ∂

∂uj to the fields Dj on Rm.Note that it is not required for Φ to be a diffeomorphism in order for (Φ∗X) to make sense asa vector field on N – as long as Φ is a smooth map such that p1 = p2 ∈ M implies Φ∗p1Xp1 =Φ∗p2Xp2 the definition (57) still makes sense. The following example gives a preview on howthis may be used in the case that the vector field has some infinitesimal symmetries as they willbe defined in the section on Lie derivatives.

Exercise 3.19 Consider M = R2 \ 0 and N = IP1. Let X(x) = (ax1 + bx2) D1|x + (cx1 +dx2) D2|x be a linear vector field on M . Define a relation ∼ on M by x ∼ y if there exists λ ∈ Rsuch that x = λy. Verify that ∼ is an equivalence relation on M .Let Φ: M 7→ N = IP1 = M/∼ be the canonical projection map which maps each x ∈ M to itsequivalence class [x] = y ∈ M : y ∼ x. Verify that if x ∼ y then Φ∗x(Xx) = Φ∗y(Xy) andhence we may Φ∗(X) does define a smooth vector field on IP1.Consider the local coordinate chart (m,U) on IP1 where U = [(x1, x2)] : x1 6= 0 and m([(x1, x2)])


is the slope of the line through the points [(x1, x2]), i.e. m([(x1, x2)]) = x2x1

. Find an explicit ex-pression for (Φ∗X)m = f(m) ∂

∂m

∣∣m

. Interpret (Φ∗X) as defining a dynamical system m = f(m)on the space of lines through the origin. In detail discuss the special cases when a = d and eitherb = c = 0 or b = −c = −1. In general relate the stationary points of Φ∗X (i.e. the zeros of f(θ)to the eigenspaces of the 2× 2-matrix with entries a, b, c and d.

Exercise 3.20 Extend the previous exercise 3.19 to a higher dimensional case. Let X(x) =∑mi,j=1 aijx

i Dj |x be a linear vector field on Rm. Use the coordinates y = (y1, . . . ym−1) on a

suitable subset U ⊆ IPm−1 = (Rm \ 0)/∼ defined by yj([x]) = xj

xm and verify that in thesecoordinates (Φ∗X) is a quadratic vector field (representing a Riccati differential equation).As an illustration explicitly write out the formula for (Φ∗X) in the case of m = 1. For funexplore the case where the matrix (aij) has a triple eigenvalue with a single Jordan block, e.g.aii = λ 6= 0, a12 = a23 = 1 and aij = 0 else. In particular, sketch the phase portrait for (Φ∗X)near y = 0 and relate it to the integral curves on IP2 (or on S2 which may be easier to visualize).

3.7 The cotangent bundle and differential one-forms

Associated to each tangent space TpM of a manifold M at a point p is a well-defined dual spacewhose elements are the linear functionals on TpM . Assembling all these dual spaces one obtainsthe cotangent bundle. Its sections, the analogues to (tangent) vector fields, are differentialforms. While such dual objects appear to be considerably less tangible to the novice, theydo have better algebraic properties than tangent vector fields. This makes them the preferredchoice in the many settings where one may choose between describing objects and propertiesusing tangent fields or cotangent fields. We begin with a brief linear algebra review.Let V be a finite dimensional vector space (over a field, here always taken to be R). A linearfunctional on V is a linear map λ : V 7→ R (i.e. λ(cv + w) = cλ(v) + λ(w) for all v, w ∈ Vand all c ∈ R). The set V ∗ of all linear functionals on V inherits a scalar multiplication andaddition from the codomain R, i.e. for linear functionals λ1, λ2 on V , c ∈ R, and v ∈ V define(cλ1 + λ2)(v) = cλ1(v) + λ2(v). It is a straightforward to check that with these operations theset V ∗ is a vector space over R.

Exercise 3.21 Suppose β = v1, . . . vm is a basis for a vector space V . Consider the mapsλj : V 7→ R defined by

λi( m∑

j=1

cjvj

)= ci where ck ∈ R. (58)

• Verify that λi ∈ V ∗.• Show that γ = λ1, . . . λm are linearly independent.• Show that every linear functional λ ∈ V ∗ is a linear combination of γ.

The exercise establishes, in particular, that V ∗ is of the same dimension as V . The basis γ forV ∗, described in this exercise, is called the dual basis to β.Novices to linear algebra often seem troubled that unlike the elements of the given vector spaceV the elements of V ∗ seem to be less tangible, and that they can be represented by an somewhatarbitrary collection of different objects. However, this drawback is easily compensated for bytheir superior algebraic properties . . . The following exercise may help a little pinning down whatthe linear functionals are (and what they are not).


Exercise 3.22 [[This is not meant to be deep, but should be fun and provide a hands-ondifferent point of view.]] Consider the vector space V of all quadratic polynomial functions onthe real line. In the usual shorthand notation V = a+ bx+ cx2 : a, b, c ∈ R.• Verify that λ1 : p 7→p(1), λ2 : p 7→p′′(23), λ3 : p 7→

∫ 10 p(t) dt, and λ4 : p 7→

∫∞−∞ e−t2p(t) dt,

are linear functionals on V .• Show that λ1, λ2, λ3 is a basis for V ∗.• Write λ4 as a linear combinations of λ1, λ2 and λ3.• Find a basis for V ∗ that is dual to the basis 1, x, x2 for V .• Find a basis for V that is dual to the point evaluations ej : p 7→ p(j) for j = 1, 2, 3.• Explain why for every fixed integer N > 0 and every fixed interval [a, b] there exist fixed

numbers αj , ξj ∈ R (not necessarily in [a, b]) such that for every polynomial function p ofdegree at most (N −1),

∫ ba p(t) dt =

∑Nj=1 αjp(ξj). (E.g. use that Vandermonde matrices

are nonsingular.)This example will be revisited in the next chapter in the context of inner product spaces.

Returning to differential geometry, define

Definition 3.8 Suppose M is a smooth manifold and p ∈ M . The cotangent space to M at p,denoted T ∗pM , is the space of all linear functionals on TpM , i.e. T ∗pM = (TpM)∗.

Recall that we defined tangent vectors Xp ∈ TpM to be linear mappings from C∞(p) to R.Turning this around we define:

Definition 3.9 For p ∈M and f ∈ C∞(p) define a map (df)p : TpM 7→ R, called the differentialof f at p, by

(df)p(Xp) = (Xpf). (59)

Exercise 3.23 Verify that for each p ∈M and each f ∈ C∞(p) the differential (df)p is a linearfunctional on TpM , i.e. (df)p ∈ T ∗pM .

Proposition 3.11 Suppose that (u, U) is a chart about p ∈Mm. Then the set (du1)p, . . . (dum)pof differentials at p is a basis for T ∗pM , dual to the basis ∂

∂u1

∣∣p, . . . ∂

∂um

∣∣p of TpM .

Proof. From the definition it is clear that

(dui)p( ∂∂uj

∣∣p) = Dj(ui u−1)|u(p) = δi

j (60)

which shows the linear independence of γ = (du1)p, . . . (dum)p. Since the cardinality of γmatches the dimension on TpM , this also establishes that γ is a basis for T ∗pM .

Note that in a chart (u, U) the coordinates ωj of any element ω =∑m

j=1 ωj (duj)p ∈ T ∗pM areimmediately obtained by evaluating ωj = ω( ∂

∂uj

∣∣p). In particular, if f, g ∈ C∞(p) are such that

for all j = 1, . . .m, ∂f∂uj

∣∣∣p

= ∂g∂uj

∣∣∣p

then (df)p = (dg)p as elements of T ∗p (M).

It is useful to compare the notion of differential forms developed here to the common usage incalculus. For illustration consider the function z = x2 +y2 (i.e, z : R2 7→ R), whose differential isdz = 2x dx+ 2y dy. Commonly dz is considered as a function of the four variables x, y, dx and


dy. Often one finds some ambiguous language that characterizes the differentials dx, dy, and dzas infinitesimal objects, yet allows the function dz to be evaluated at a point like (x, y, dx, dy) =(2, 3, 0.2,−0.1). Thus dz is now considered as a function dz : R4 7→ R. It is apparent that(2, 3, 0.2,−0.1) denotes (are the coordinates of) the (infinitesimal?) tangent vector (0.2,−0.1)at (2, 3). In our notation this tangent vector is written as 0.2 ∂

∂x |(2,3)−0.1 ∂∂y |(2,3). Note that it is

consistent with our language to use dx and dy as coordinates in the tangent plane – we merelymay regard them as linear functions, here on T(2,3)R2. On manifolds, we clearly distinguishbetween (dx)p and (dx)q at different points (just as we associate tangent vectors to fixed points).In particular, dx = 0.2 is simply a shorthand for (dx)(2,3)(0.2 ∂

∂x |(2,3) − 0.1 ∂∂y |(2,3)) = 0.2.

Indeed, with differential forms we now alternatively may express a tangent vector Xp ∈ TpM ina chart (u, U) about p as

Xp =m∑

j=1

(Xpuj) ∂

∂uj

∣∣p

or Xp =m∑

j=1

(duj)p(Xp) ∂∂uj

∣∣p

(61)

and (duj)p, j = 1, . . .m are legitimate coordinate functions, or simply “coordinates” (?) oftangent vectors.In complete analogy to the tangent bundle assemble all cotangent spaces T ∗pM into the cotangentbundle, denoted T ∗M . It is a vector bundle over M with bundle projection again denoted by π.For any chart (u, U) of M define U = π−1(U), and, u : U 7→ R2m by

u(p, ωp) = (u1(p), . . . , um(p), ωp( ∂∂u1

∣∣p), . . . , ωp( ∂

∂um

∣∣p)). (62)

Using proposition 3.11 it is clear that u is a bijection onto its image. As in the case of TM , it ispossible to equip T ∗M with a topology such that the maps u are homeomorphisms (onto theirrespective images). In more technical work one may show that the topology is metrizable, andvia the next exercise, T ∗M is a smooth manifold.

Exercise 3.24 Suppose (u, U) and (v, V ) are charts on M , and (u, U), (v, V ) are defined asabove. Verify that the transition maps v u−1 : u

(U ∩ V

)7→ v

(U ∩ V

)⊆ R2m are smooth

bijections between subsets of Euclidean spaces.

Definition 3.10 The (smooth) sections of the cotangent bundle, that is, the (smooth) functionsω : M 7→ T ∗M satisfying π ω = idM are called (smooth) differential one-forms. The space ofall smooth differential one-forms on M is denoted by Ω1(M).

As a map between smooth manifolds, a section ω : M 7→ T ∗M is smooth if for all coordinatecharts (u, U) of M and (v, V ) of T ∗M the maps v ω u−1 : u (U ∩ V ) 7→ v

(U ∩ V

)⊆ R2m are

smooth as maps from subsets of Rm to subsets of R2m.In particular, in a chart (u, U) a differential one-form ω ∈ Ω1(M) may be written as a linearcombination ω =

∑mj=1 ωj du

j with smooth functions ωj ∈ C∞(M) defined by ωj = ω( ∂∂uj ).

For every function f ∈ C∞(U) defined on an open subset U ⊆ M define a smooth differentialone-form df ∈ Ω1(U) by (df)(p) = (df)p for p ∈ U . (More pedantic writers may prefer (df)(p) =(p, (df)p).) In a coordinate chart (u, U) this becomes df =

∑mj=1

∂f∂uj du

j .

After this brief discussion of smoothness and local representations we take a look at the algebraicstructure and properties of Ω1(M). We already have routinely combined smooth functions f, g ∈C∞(M) and differential forms ω, η ∈ Ω1(M) to write e.g. (fω + gη). A brief reflection shows,


that this is permissible, and indeed yields new differential forms in Ω1(M): Indeed any T ∗pM is aR-vector space and for any Xp ∈ TpM we interpret (fω+ gη)p(Xp) as f(p)ωp(Xp)+ g(p)ηp(Xp).The next exercise addresses the smoothness.

Exercise 3.25 Suppose f, g ∈ C∞(M) and ω, η ∈ Ω1(M). Argue (from the definition of smooth-ness of maps between manifolds) why fω + gη is indeed a smooth differential form on M .

It is a straightforward to verify that the usual mixed associative and mixed distributive lawshold, and thus ω ∈ Ω1(M) has the structure of a C∞(M) module.On the other hand, every differential one-form ω ∈ Ω1(M) is naturally also a functional thatmaps ω : Γ∞(M) 7→ C∞(M), defined pointwise by ω(X)(p) = ωp(Xp). To verify that ω(X) isindeed a smooth map locally expand ω(X) in a coordinate chart (u, U)

ω(X) = (m∑

i=1

ωi dui)(

m∑j=1

(Xuj) ∂∂uj ) =

m∑i=1

ωi · (Xui) (63)

and use that ωi and (Xui) are smooth functions since ω and X are a smooth differential formand a smooth vector field, respectively.Moreover, one readily observes that if f ∈ C∞(M) (in a chart (u, U)

ω(fX) =m∑

i=1

ωi dui (

m∑j=1

(fXuj) ∂∂uj ) =

m∑i=1

f · ωi ·Xui = f

m∑i=1

ωi dui (

m∑j=1

(Xuj) ∂∂uj ) = fω(X)

(64)establishing that any ω ∈ Ω1(M) is not only an R-linear map, but indeed a C∞(M)-linear mapfrom ω ∈ Γ∞(M) to C∞(M), written

Ω1(M) ⊆ HomC∞(M)(Γ∞(M), C∞(M)). (65)

3.8 Cotangent maps and pullbacks of differential forms

The next step is to analyze the analogues of the tangent maps associated to a smooth functionbetween manifolds. Recall from linear algebra that every linear map φ : V 7→W between vectorspaces induces a dual map φ∗ : W ∗ 7→ V ∗, defined by (φ∗λ)(v) = (λ φ)(v) for v ∈ V andλ ∈W ∗.

Definition 3.11 Suppose Φ ∈ C∞(M,N) is a smooth map and p ∈ M . Define the cotangentmap Φ∗p : T ∗Φ(p)N 7→ T ∗pM as the dual of the tangent map Φ∗p, i.e. Φ∗p = (Φ∗p)

∗.

Note that this means if ωΦ(p) ∈ T ∗Φ(p)N and Xp ∈ TpM then(Φ∗p ωΦ(p)

)(Xp) = ωΦ(p) (Φ∗pXp) . (66)

Exercise 3.26 Let Φ ∈ C∞(M,N), Ψ ∈ C∞(N,P ), and p ∈M . Verify (Ψ Φ)∗p = Φ∗p Ψ∗Φ(p).

Unlike the situation of the tangent bundle it is in general not possible to combine all maps Φ∗p,p ∈M together to get a well-defined map from T ∗N to T ∗M . Indeed, the first hint at problems isthat the maps Φ∗p are naturally indexed not by their domains but by their codomains! Indeed, ifp, q ∈M are such that z = Φ(p) = Φ(q) ∈ N then there are well-defined maps Φ∗p : TN 7→ T ∗pM ,


Φ∗q : TN 7→ T ∗q M with the same domain, but different codomains (unless this implies p = q, i.e.,unless Φ is one-to-one). Nonetheless, in the case that Φ is one-to-one (i.e. especially if Φ is adiffeomorphism) define Φ∗ : T ∗N 7→ T ∗M pointwise by Φ∗(ωΦ(p)) = Φ∗p(ωΦ(p)) for p ∈ M andωΦ(p) ∈ T ∗Φ(p)N .This lack of well-defined cotangent maps between cotangent bundles is a small price to pay fornow being able to map sections: Recall, that in general it is not possible to map a vector fieldX : M 7→ TM forward to a vector field Φ∗X : N 7→ TN . However, it is always possible to pullback differential forms (along smooth maps):

Definition 3.12 If Φ ∈ C∞(M,N) and ω ∈ Ω1(N) define the pullback Φ∗ω : M 7→ T ∗M of ωby Φ to M for p ∈M by

(Φ∗ω)(Xp) = ωΦ(p)(Φ∗pXp). (67)

Exercise 3.27 Suppose Φ ∈ C∞(M,N) and ω ∈Ω1(N). Verify directly that (Φ∗ω) ∈ Ω1(M),i.e. that Φ∗ω is smooth.

This is a good place to comment about some unfortunate terminology. Associated to a mapΦ: M 7→ N are two maps, Φ∗ : TM 7→ TN , going in the same direction, and Φ∗ : T ∗N 7→ T ∗M ,going in the opposite direction. Modern language would use the attribute covariant for the first,and the attribute contravariant for the latter. Unfortunately, classical language used the samewords for co-tangent and tangent vector fields. Quoting from Spivak vol.I, p.156 “. . . and no onehad the gall or authority to reverse terminology so sanctioned by years of usage. So it’s veryeasy to remember which kind of vector field is covariant, and which is contravariant – it’s justthe opposite of what it logically ought to be. (I.e. sections X : M 7→ TM are called contravariantvector fields, and sections ω : M 7→ T ∗M are called covariant vector fields . . . )Pullbacks of cotangent vector fields are especially useful when working with imbedded subman-ifolds. More specifically, suppose that M ⊆ N is a submanifold and consider the inclusion mapı : M → N . Then every differential form ω ∈ Ω1(N) immediately gives rise to a differential formı∗(ω) ∈ Ω1(M). Indeed, this is used so often that one routinely even uses the same symbol ωfor ı∗(ω). On the side note that there is no equivalent to this for tangent vector fields: Indeed,for any vector field X ∈ Γ∞(M) there are in general many extensions to a vector field on N .Conversely if N ⊆ M is a submanifold of positive codimension and Φ ∈ C∞(M,N) then Φis necessarily many-to-one and unless something special happens there is little hope that thecollection of tangent vectors Φ∗pXp (with p ∈M) are the image of a vector field on N .

In practical examples one routinely needs to calculate the pullbacks of differential forms in termsof local coordinates. Thus consider a smooth map Φ ∈ C∞(M,N), local and coordinate charts(u, U) about a point p ∈M and (v, V ) about Φ(p) ∈ N . Due to the linearity of Φ∗p is suffices toconsider the pullbacks Φ∗(dvi). As an immediate consequence of the earlier calculations (3.3) ofthe tangent map in coordinates find

(Φ∗p(dv

i))

∂∂uj

∣∣p

= dvi(

Φ∗p ∂∂uj

∣∣p

)= dvi

( n∑`=1

∂(v`Φ)∂uj

∣∣∣p· ∂

∂v`

∣∣Φ(p)

)= ∂(viΦ)

∂uj (68)

and consequently for ωi ∈ C∞(N)

Φ∗( n∑

i=1

ωi dvi)

=m∑

j=1

( n∑i=1

ωi∂(viΦ)

∂uj

)· duj . (69)


As expected this means that the coordinates transform by matrix-multiplication. One maylook at this in different ways: If assembling the coordinates ωi into column vectors then thecoordinates of the image are obtained by left multiplication by the transpose of the usual Jacobianmatrix with components ∂(viΦ)

∂uj . A more elegant way to interpret the sum in equation (69) isin terms of right multiplication of row vectors by the standard Jacobian matrix – no transpose.Thus if we write a = (ω1, . . . , ωn) and b = (du1(Φ∗ω), . . . , du1(Φ∗ω)) then b = aC where C isthe matrix with components Cij = ∂(viΦ)

∂uj .Consistently using this convention of representing (in local coordinates) tangent vector fields bycolumn vectors and differential forms by row vectors facilitates many calculations. In particular,the evaluation of a differential form on a tangent vector field becomes in coordinates simply thematrix product of a row vector with a column vector (in this order). Moreover, the definingequation (Φ∗ω)Xp = ω(Φ∗Xp) is simply interpreted as associativity of matrix multiplication:Let, as before, a = (ω1, . . . , ωn) denote the coordinates of a differential form ω on N , C theJacobian matrix with components Cij = ∂(viΦ)

∂uj , and let now ξ = (Xpu1, . . . , Xpu

m)T denotethe column vector of the u-coordinates of the tangent vector Xp ∈ TpM . Then we simply have

(Φ∗ω)Xp = ω(Φ∗Xp) ←→ (aC)ξ = a(Cξ). (70)

Formally, it is at times convenient to assemble the basis vectors into formal row and columnvectors. To be consistent introduce the formal column vectors α = (Φ∗dv1, . . . ,Φ∗dvn)T andβ = (du1, . . . dum)T . Then α = Cβ from (68). Together with the notation of the previousparagraph, this provides for such nice shorthand notation as

Φ∗ω = aα = a(Cβ) = (aC)β = bβ. (71)

Exercise 3.28 Suppose Φ ∈ C∞(Mm, Nn) and Ψ ∈ C∞(Nn, P r) are smooth maps betweenmanifolds, p ∈ M and Xp ∈ TpM . Furthermore, suppose (u, U), (v, V ) and (w,W ) are localcoordinate charts about p ∈ M , Φ(p) ∈ N and (Ψ Φ)(p) ∈ P , respectively. Verify that thematrix representing (Ψ Φ)∗p with respect to (u, U) and (w,W ) is the product of the matricesrepresenting Φ∗p (with respect to (u,U) and (v, V )) and Ψ∗

Φ(p) (with respect to (v, V ) and (w,W )).

Exercise 3.29 Revisit the exercise 3.17 with M = x ∈ R2 : x2 > 0 equipped with rectangularcoordinates (x, y), polar coordinates (r, θ), and the mix (ξ, ρ) defined by ξ = x and ρ = r.For each pair of coordinates calculate the Jacobian matrix C with components Cij = ∂(viΦ)

∂uj ,when Φ = idM is the identity map, and use this to write each set of basic differential formsdu1, du2 as a linear combination of each other set dv1, dv2. In particular sketch these basicco-tangent vector fields using arrows . . . . Compare to the pictures for the basic tangent vectorfields ∂

∂u1 ,∂

∂u2 from exercise 3.17.

Exercise 3.30 Consider the imbedded sphere S2 ⊆ R3 (i.e. M = S2, N = R3 and the inclusionmap Φ = ı : S2 7→ R3) and the standard spherical coordinates (u, U) = ((θ, φ), U), e.g. withU = (θ, φ)−1(−π, π) × (0, π)) on M and the Cartesian coordinates (v, V ) = ((x1, x2, x3),R3)on N .Explicitly calculate the pullbacks ı∗dvi for i = 1, 2, 3 in terms of du1 = dθ and du2 = dφ.Locate all points p ∈ M where any of these cotangent vector fields vanish. Describe the vectorfields pictorially, both as arrows on the sphere, and as arrows on (−π, π) × (0, π) (technically,this means sketching the vector fields

(u−1

)∗ ı∗dvj).


In a subsequent section we will return to differential forms to investigate when a differential formω is the differential of a smooth function. which will lead to powerful integrability theorems. Thereader is encouraged to continue comparing and contrasting the algebraic ease of working withdifferential form and the more tangible, visual aspects of tangent vector fields.

3 the tangent bundle - arizona state university

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