IC/77/1

HTEEHHAL EEPOET(Limited distribution)

International Atomic Energy Agency

and

United Nations Educational Scientific and Cultural Organization

INTERNATIONAL CENTRE FOR THEORETICAL PHYSICS

SPACETIME EDGE GEOMEXBI •

(Chapters I 4 I I )

C.T.J. Dodson • •

In ternat ional Centre for Theoretical Physics, T r i e s t e , I t a l y ,

MIRAHAEE - TRIESTE

January 1977

• To be submitted for publication

** On sabbat ical leave from the Department of Mathemrtlca, University of

Lancaster, England.

COHTEHTS

ABSTRACT

This monograph i s intended to be a mathematically rigorous account ofthe current position on the 'bundle-completion of spaeetimes in general re la t iv-i ty ; Borne new material i s included. I t arise3 from work done at the Inter-national Centre for Theoretical Physics, and lectures given there at the in-vitation of Professor Abdus Salsm during 19T6/T. The text is reasonably self-contained in that the material on Lie groups and fibre bundles likely to beunfamiliar to some of the intended readership is reviewed with many examplesin a preliminary chapter. An elementary knowledge of d i f fe rent ia te manifoldsand maps among them is assumed. The subject matter is geometrical singular-i t i e s and the approach here is intended to promote geometrical intuition with-out sacrificing rigour.

ACKNOWLEDGMENTS

I. BACKGBOUHD WITH EXAMPLES

1. TOPOLOGY

1.1 Separation: T , T , T,, spaces

1.2 Subspace topology

1.3 Semimetric topology

1.1* Uniqueness of limits in !„

1.5 Agreement of functions

1.6 Graph of functions

1.7 Regular, normal, T,, T. spaces

1.8 Regularity of semimetrics

1.9 Uniform continuity

1.10 Unique uniformly continuous extension

1.11 Equivalence of metrics

1.12 Topological boundary of a set

1.13 Covering space

2. MANIFOLDS

2.1 Vector field

2.2 Lie bracket

2.3 Integral curve2.1* Riemannian distance; Cauchy completion

2.5 Geodesic completeness

Toe author would like to thank Professor Abdus Salam, theInternational Atomic Energy Agency and UNESCO for hospitality at theInternational Centre for Theoretical Physics, Trieste. He would also liketo thank The Royal Society for the award of a European Science ExchangeProgramme Fellowship for the year 1976/TT.

3. LIE GROUPS

3.1 Definition. GH(n;R)3.2 Left invariant vector field3.3 Lie algebra; subalgebra. %l(n;El)3.1* Adjoint representation: ad and Ad * D(ad)

3.5 One-parameter subgroup. Exponential map3.6 Right action of G on manifold P3.7 One-parameter subgroup G for y ( }

3.8 Transitive, free, effective actions

3.9 Orbit of action. Quotient space P/G . Connected component G

-i--ii-

IT

u. FIBRE BUNDLES

l*.l P r i n c i p a l f i b re bundle (P,G,M) . S t ruc ture (i^oup. Frame bundle LM

It. 2 Fundamental vec tor f i e l d v* for Y * S

I*.3 Associated bundle . Tangent, cotangent and t enso r 'bundles. F ie lds

It. It V e r t i c a l subspace, G £ T^P

It. 5 Canonical 1-fonn 8 : TLM + En

1).6 Universal covering manifold. Homotopic curves

5

5.1

5.2

5.3

5.It

5.55.6

5.7

5.8

5.95.10

5.11

5.12

5.13

5.ll»

COHMECTIONS

Definition. H 9 0u uHorizontal l if t of vector fields: v »

T Pu Horizontal subspace H

*°(u)

V as derivationProperties of horizontal l i f tHorizontal l i f t of curves: c ** c* . Parallel transport T .Geodesic curves. Standard horizontal vector fieldConnection form tti : TP * 4 . Coordinate expressionsLoop space. Holonomy groups of 7 : *(u) andHolonoay bundle P(u) . Equivalence -"»Levi-Civlta connectionBundle of orthonoraal frames CMExponential map

Riemannian completeness properties

Conditions for completeness

Curvature. Rlemann tensor

II. COBfflgCTIOS GEOMETRY FROM FRAME BUHDLES

1 . SCHMIDT'S BUHDLB METRIC

1.1 Existence

1.2 Uniqueness

1.3 Uniform ac t ion of Gl(n ; l t ) ; unique extension t o completion

l .U The b-boundary, SH • L ' M / G * \ H

2. THE METRIC TOPOLOGY OF L'H

2.1 <J acts transitively on fibres

2.2 H£, 1* an open map2.3 Completeness of fibres vith the Induced metric

-iii-

2.k Fibres in L'M are homogeneous spaces

2.5 Boundary points generated by horiaontal Cauchy sequences3.6 Holonoay bundles generate the same b-boundary

3 . THE QUOTIENT TOPOLOGY OF M

3.1 The semimetrie structure, p

3.2 If L'M is complete and p a metric then M 1B complete3.3 M is not T, If the orbits of G are not closed3.It M is T <* graph G is closed3.5 Incomplete fibres in the full metric

3.6 M is geod«Bleally complete i f L'M is complete

3.7 M is homeomorphic to the Cauchy completion in Riemannian case3.8 M need not be locally compact3.9 b-incompleteness of curves in M3.10 Imprisoned b-incompleteness

3.11 A b-boundary contains the topologisal boundary3.12 M is ^-complete i f L'M is complete

3.13 Projection of finite incomplete curves

-iv-

REFERENCES

1) BoBshard, B . , Commun. Math. Phys . U6, 2 6 3 - 8 ( 1 9 7 6 ) .

2) Biahop, R.L. and Crittenden, R.J., Geometry of Manifolds (Academic Press,

Hew York 196U).

3) Clarke, C.J.S. , Commun. Math- Phys. 1*1, 65-78 (1975).

k) Dieudonne, J . , Foundations of Modern Analysis (Academic Press, Hew York

I960).

5) Dodson, C.T.J., ICTP, Trieste, Internal Report IC/76/112 (1976),

submitted to GB8 Symposium, Waterloo, Ontario, Canada, August 1977-

6) Dodaon, C.T.J. and Poston, T., Tensor Geometry (Pitman, London 1977).

7) Dodson, C.T.J. and Sulley, L.J. , Lett. Math. Fhya. (in press).

8) Duncan, D.P. and Shepley, L.C., Huovo Cimento 2bB, 1, 130-U (1971*).

9) Duncan, D.P. and Shepley, L.C., J. Math. Phys. 16, 3 , U85-l*92 (1975).

10) Be l l s , J . , "Fibre bundles", pp.53-82 in Global Analysis sad I t s Applications

Vol.1 (IAEA, Vienna 1971*).

11) Ehresmann, C , Colloque de Topologie, Bruxelles, 5-8 June 1950 (CBKH).

12) Garoch, R.P., Ann. Phys. US, 526-5kO (1968).

13) E&Jicek, P. and Schmidt, B.G., Commun. Math. Physs. 23, 235-295 (1971).

Ik) Hawking, S.W., pp.275-9 in Proceed^ga of Liverpool Singularities

Symposium I I . Lecture Notea in Mathematics 209 (Springer, Berlin 1971).

15) Hawking, S.W. and Ki l l s , G.F.B., The Large Scale Structure of Space-time

(C.U.P., Cambridge 1973).

16) Ihrig, E . , Gen. Relativ. Gravitation, 7, 3 , 313-323 (1976).

17) Kelley, J .L . , General Topolopy tvan Hostrand, Pvinceton

18) Kobayashi, S. and Homizu, K., Foundations of Differential Geometry.

Vol.1 {Interscienee, Hev York 1963).

19) Pontryagin.L.S., Topolotitlcal Groups. 2nd ed. (Gordon and Breach, Hew York

1966).

20) Sachs, B.K., Commun. Math. Phya. 33, 215-220 (1973).

21) Schmidt, B.G., Gen. Relativ. Gravitation, 1 ,3 , 269-280 (1971).

22) Schmidt, B.O., Commun. Math. Phys. 29, t9-5U (1973).

23) Schmidt, B.G., Commun. Math. Phys. 36, 73-90 (197M.

2h) Steenrod, H.E., Topology of Fibre Bundles (Princeton University Press,

Princeton 1951).

- v i -

, v \-

I.

I. BACKGROUND WITH EXAMPLES

The material which it is convenient to have available for reference in

the sequel we gather under five section headings:

1. Topology

2. Manifolds

3. Lie groups

4. Fibre bundles5. Connections

For the topology, which mainly pert&ins to separation and convergence

properties, conveniently readable and widespread references are Kelley [17]

and Dieudonne [k]. Little in the way of amplification seems necessary for that

section.

The differential geometry is covered, though rather tersely for physicists,

in Kobayashi and Nomizu ElS]. The text by Bishop and Crittenden [2] deals with

similar material but contains more breathing spaces in its exposition and more

actively encourages an intuitive feel for the subject. A geometrically

motivated account of manifolds, their curvature and connections is given in

Dodson and Poston [6], which begins with vector spaces and proceeds to a rigor-

ous formulation of relativistic spacetime. Our notation will follow this text

when it is convenient, otherwise it will be similar to that in the former hooks.

The end or absence of a proof vill be signified by • .

1. TOPOLOGY

By the term "space" we always mean a topological space, perhaps alsohaving additional structure that is plain from the context. A map betweenspaces, denoted by f say, will be represented as

f : X + Y : x«- f(x)

vhen f has domain X and i t sends the arbitrary element x (, X to f(x) t Y.Moreover, we shall use the notation

f+ : sub Y •+ sub X : B "• {i * X|f(x) * B}

for the induced map on subsets of Y vhen the image of f is Y . Thus we

reserve the symbol f for the case vhen f is a bisection and t is

its inverse. The identity map on any space X is denoted I .

-1-

1.1.10l.i A space is called:

1.2

1.3

l.k

1.5

1.6

1.7

1.8

1.9

1.10

(i) T if for each pair of distinct elements from it there is a

neighbourhood of one to which the other does not belong;

(ii) T if each set which consists of a single point is closed;

(iii) T or Hausdorff if whenever x and y are distinct elements

from it there exist disjoint neighbourhoods of x and y .

Hence metric spaces are always T spaces.

Every subspaee of a T [respectively, T ] space is a T [respect-

ively, T g ] space. •

For any semimetric space (X,p) we have the equivalences: X is T

«=• X is T <& p is a metric. •

Every convergent sequence in a T space has a unique limit. •

Let g,f : X -+ Y be continuous maps with Y a T_ space. Then

{x £ x|f(x) = g(x)} is closed and if it is also dense in X then

f = g . •

Let f : X •+ Y be continuous with X a !„ space. Then graph

f • {(x,f(x))« X x Y} is closed in the product topology and f {y)

is closed in X for all y t Y , •

A space is called:

(i) regular if for any subset A and point x * A there exist

(ii)

disjoint open sets containing A and x ;

T i f regular and

( i i i ) normal i f given any two disjoint closed subsets there are dis-

joint open subsets containing them;

(iv) TV if normal and T .

Every semimetric space is regular and normal but not necessarily

Hausdorff. •

A map f : X •* Y between metric spaces (X,d) and (Y,d' } i s

called uniformly continuous i f ( Ve > 0)( 3 6 > 0) : d(x,y) < S ^

d ' ( f (x) , f (y) ) < £ .

Let A be a dense subset of a metric space (X,d) and f a uniformly

continuous map of A into a complete metric space (Y.d1) . Then

there exists a uniformly continuous map f : X + Y , coinciding with

f in A . •

-2-

1.1.13

1.11 Let (X,d) , (X,d*) be metric spaces having the same underlying space.

Then we have the following:

i) If IJJ is a homeomorphism then the d-topology coincides with the

d'-topology for X ,

i i ) If Ix is uniformly continuous (with a unifornly continuous inverse)

between (X,d) and (X.d1} then the Cauchy sequences agree for d

and d1 . •

In ease i ) we call d and d' topologieally equivalent distances

ana in case i i ) we call them uniformly equivalent: from 1.9 the latter implies

the former.

1.12 Let A be a subset of a topological space X . We shall use the follow-

ing notation:

i) lnt(A) for the interior of A , the largest open subset of A ;

i i ) A for the closure of A , the smallest closed set containing A ;

i i i ) A for the topological boundary of A , the set AC fi (j \A)C .

We recall a few properties:

a) A is open • * • int(A) = A <=> A C\ A = 0 ;

l ) A i t closed * * A £ A •*+• A contains i t s limit points;

c) ( | \A) C = S \ int(A) ;

d) Ae s int(A) = A ;

e) A U A • A° ;

f) A x A = int(A} . •

1.13 A covering apace of a connected, locally arcwise connected topological

apace X , is a connected space E such that:

i) there exists a projection p : E •* X ;

ii) for all connected open neighbourhoods U of any x * X , each

connected component p (U)' of p (U) is open in E and homeo-

morphic to U by the restriction of p : p (U) = U .

A covering space is called a universal covering apace if it is

simply connected.

If X is a manifold then every covering space has a unique mani-

fold structure which makes the projection differentiate {cl.k.i). •

-3-

2.

1.2

Two covering spaces p : E - * - X , p ' : E ' - * - X are isomorphic if

there exists a homeomorphism f : E - E' such that p'«f » p .

Note that we use = to denote that two spaces axe homeomorphic,

that is topologieally equivalent. We shall use s to denote a

stronger equivalence that is of interest when additional structure is

present in the spaces. For example, in the case of metric spaces

(X,d) , (Y,d') the existence of a homeomorphism (X = Y) that is

also an isometry will be indicated by X = Y . Similarly in the case

of manifoJds M , M' (cf. 2 below) if their underlying topological

spaces are homeomorphic (M ~ M') via a map f that is also a

diffeomorphisia (f and t are continuously differentiable) then we

write M 2 M1 .

MAMIF01DS

By t h e term "manifold" we s h a l l mean a f i n i t e - d i m e n s i o n a l , smooth r e a l

manifold . For t h e s t anda rd manifolds l i k e R , Rn and S and t h e i r p roduc t s

we shall assume atlases derived from the identity map on the relevant IP

that contains them. We recall that every point x in a topoloBical n-

manifold M has a neighbourhood homeomorphic to an open set in K . For

a smooth n-manifold M , each such x also possesses a tangent vector space

T M ; these tangent spaces can be collected to form a smooth 2n-manifold, the

tangent bundle TM , with underlying set \) T M and canonical projectionx « M

- I t -

1 . 2 . 1

TM •* M send ing T M t o x fo r a l l x £ M

Each of t h e t a n g e n t spaces T M i s isomorpli ic - o B and so we s h a l l o f t en

denote an element in T M by a pair (x,v) with v in some convenient iso-

morph of Rn , induced by a choice of basis for TM .

The differentiable structure of a smooth manifold allows us to define

differentiabili ty for maps between manifolds. In particular every point of

such an n-manifold lias a neighbourhood that is dlffeomorphic (denoted by £ )

to an open set in RQ . Suppose that M and M_ are smooth manifolds of

dimensions n, and , respect ively .

f :

Given any different iable map

2

there i s induced a map, i t s d i f f e r en t i a l ,

Df : TM •* TM, ,

which for a l l x t M has a l inear r e s t r i c t i on

V : TxMl * Tf(x)M2 "If f and 1 are local chart maps about x and f(x) then we have a localrepresentation of f (with domain contained in Rn signified by >-• )

, -1 Rn ~ S m : (x1) »

The Jacobian matrix (f?) at (x1) =f(x) induces the local representation

of the derivative D f by *)

D f : Pn-x R" : (v ) '

We shall find it convenient to agree to write

(f(x).f^ v1)

vhea a choice of charts has been decided.

2.1 A vector field on a manifold M is a map

v : 11+™

such that for all x « M , II^vU)) = x . (Cf. U.3 Example 3.) We shall

always assume that our vector fields are smooth. An equivalent definition

is that w is a smooth section of the surjection Hj, • The set of such will

be denoted T M ; under pointvise operations it becomes a vector space (infinite-

dimensionaj).

^ We use Einstein's summation convention throughout.

- 5 -

1.2.5

It tujrns out that vector fields are derivations on smooth ree.1 functions

on the manifold.

£ . 2

2.3

2.1+

2.5

, w on a manifold M i s the

and defined to act on al l smoothThe Lie bracket of two vector fields

unique vector field dencted [v,w]

f : M-ffi by

[v,w](f) = v(w(f)) - v(v(f)) . •

Jin integral curve of a vector field w on M is a curve c : t •-* c(t)

in M such that its tangent vector c(t) at each t satisfies

c(t) = w»c(t) . Such curves always exist for smooth vector fields

and they are essentially unique (cf. [6] for a recent geometrical

proof). •

Suppose that (M,g) is a connected Riemannian manifold,

defines a distance function.Then g

d_ : M x M •+ E : (x,y) "• inf{ e £ T(x,y)}

where r (x ,y ) is the class of a l l piecewise continuously differentiable

curves from x to y in M . Thus we have:

i) (M,d-) is a topalogical metric space and the metric topology co-

incides with the manifold topology.

i i ) (M,g) is called complete if (M,d,) is a complete metric space.S

A Riemannian manifold that is not complete admits a completion asa metric space by the standard procedure using equivalence classesof Cauchy sequences. We shall denote the extension of a metricd to this completion by d . •

Example. The point we illuminate is that if (M,g) is not completethen for^some x,y t H there may be no c e r{x,y) such that the

el of c equals the distance d_(x,y) , Takelength I | |o | of c equals the distance d_(x,y) ,2 J E

M = R \ {(0,0)} with the Euclidean metric and choose x = (-1,0) ,y = (1,0} , Then plainly d.(x,y) = 2 but there is no curve of thislength joining x and y , in M ._ 2M of M , is easily seen to be E

In this case the Cauchy completion

A connected Riemannian manifold turns out to be complete if and only

if every geodesic can be extended to arbi trar i ly large parameter

values; however, this correctly belongs in Sec.5,under connections.

(Cf. 5.8, 5.11 and 5'. 12.) •

-6-

1.3.2

3. LIE GROUPS

Global algebraic actions on manifolds prove to be very powerful in-

vestigative tools as well as providing elegant geometrical constructions. We

are Interested in the actions of groups on smooth manifolds, so we shall want

the groups to be smooth themselves with smooth actions. We are forced to the

following definition.

3.1 A Lie group is a group that is also a smooth manifold *uch that the

group operation (a,b) >-* ab is smooth.

tiote that (a,b) •* ab" being smooth is simply a short way of

saying that the following maps are smooth for all a,b in the group:

L ; b » ab

H

-1

b •

& •

• ba

-1

Example 1 The general linear group G£(n;R) of a l l n x n real non-^ 2

singular matrices; i t forms an open submanifold of E . In particular

Gl(l;B) = R Mo} vhich we shall denote by B* .

Example 2 The group of matrices

with 1 > v i fi

This is important in relativity where it generates a one parameter

subgroup of the Lorectz group. It has an equivalent representation

coshx

Binhx

sinhx

coshxwith x *

3.2 A left invariant vector field of a Lie group G is a vector field

w : G •* TG that is fixed under the differentials of left translations.

This means that for all g * G , L : G + G : h - + g h , left trans-

lation by g , with the differential DL : TG •* TG has the property

that

DLg w(h) = vtL (h)) = w(gh) , V h t G .

Example. Let G » GJt£n;R) . Then for any a t G ,

-7-

3.3

3.4

1 5 = (ta.A) | A is an n x n real matrix} .a

In particular for n = 1 we have G = R* and for all g t E*

L : R* -> R* : a — ga

DaLg : T a B * - T g a R » : (a,A) ~ (ga.gA) .

The left Invariant vector fields of a Lie group G form a vector space,

and an algebra under the Lie bracket composition called the Lie algebra

i of a .

As a vector space we have an isomorphism of "5 with the tangent

space to the identity e t G

i + T G : Y ** y(e) •

Hence 3 has the same dimension as the manifold G .

As an algebra we find that the inclusion of ^ in X (G) , the

Lie algebra of a l l vector fields on G , is a homomorphism so *i is

a Lie subalgebra of )t (G) . •

Example• For G = G£(n;R) we have "4 = iJKn;R) consisting essentially

of a l l n x n real matrices A , which determine fields by

A : G + TG : a * U,A) * T G .

The Lie bracket operation is the composite matrix product

[A,B] = AB - HA .

We pursue the one-dimensional example in 3-2 to display the left

invariance property. Plainly we have 4i(l;R) * H . Let y t R .

Then we require

Y : fi» i- TR» : a H- («.Ya)

such that for a l l a,g e E*

DL Y(a) = Y(L (a)) .ft S

Hence at the Identity e • 1

gy(e) =

So we have y{a) = aY(l) = aY

ytg)

say.

The adjoint representation Ad , of a Lie group S In i t s Lie algebra

*i i s obtained from the automorphisms

ad(g) = L P. ± ; G + G : h «• ghg"1 , Vg £ G ,S o

-6 -

by putting

AdCg) •- i i i

1.3.5

-i)Y

This of course implies that left invariance is preserved. •

Example. The adjoint representation is trivial for any g t R* since

L I ]_ : B» + B» : h »*

so Ad(g)

3.5 Every y in i the Lie algebra of a Lie group G generates a one-

subgroup of G , as follows.

Let c : t •* y be the integral curve in G (of. 2.3), determined

for |t| < £ for some real £ > 0 , with initial conditions:

Y o - c (0) = e the identity in G and c (0) = y(e) t TJ2 (cf. 3.3).

Define the map

<pt : a •* G : L (y t)6

t < E

How this is valid for al l g * G and hence admits an extension to all

t * R . It turns out that we have a group because for al l g fc G

Hence the required subgroup of G is

We also obtain the exponential map

exp : -• G : y »•

Example. Por G = G£{n;R) we find that the exponential map coincides

with the usual exponential function for matrices y ,

k = 0

We can see something of why this is so from the case n = 1 . We

seek the integral curve for Y * *it(l;S) = R given by (cf. example 3.3)

cy : C-e.e) - R»

subject to c (0) = e = 1 and c (0) = Y(I) = Yo

and subject at all t » (-e,e) to

-9-

0 i, (-e,s)

1.3.7

cyCt) = Y

So our differential equation is

3-6

3.7

and the required solution is

c : (-e,e) •+ fi* : t * e

Finally, the one-parameter subgroup is given by

Vf, : R* •+ E* : g •* ge

and the exponential map i s

e x p : B •+ K* : y •+• e •

Let G be a Lie group and P a smooth manifold. Then G acts on

P to the (or, on the) right if there is a smooth map

P x G + P : (u,g) •* R (u)

satisfying

i ) g : P ->• P : u "• H (u) i s a diffeomorphism, V g e G ,s

ii) R . (u) = S. (H (u)) , »rfg,h « G , V u t P .gh h g

2Example 1. Take G = R* and P = (B with

fi x R » + H . (Cx,y),g) r* (gx,gy) .

Example £. Take P = G and use right translation in G :

G x G -<- G : Ch,g) •*• fl (h) = hg .

Let a Lie group G act on the right of a manifold P . Then, from3.5, every y t *( determines a one-parameter subgroup of G

G = {<F (e) = exp ty | t t K}7 t

vhich acts on the right of P by its inclusion in G ; it also has

the following properties:

i) There is through each u t P a smooth curve

vith tangent vector y* « T P at t = 0 (ct.h.2).

-10-

1.3.8

i i ) Denote by 3fP the Lie algebra of vector fields on P and definea map

where

Y * : P •+• T P : u *

exp

Then $ is a Lie algebra homomorphism. If the only element of

G with a fixed point i s e , then Y* i a n o t t I l e z e r o vector at

any u t P , for non-zero Y * "4 • •

Example 1. When G = 0(2;R) , the orthogonal subgroup of G((2;R) ,then for any

[ 0 6Y -

L - e owe have

cos 6 sine]

[-sin6 cosej

Example 2. For 0 = GJl(l;R) = R* we have from 3.5 for any

Y * 5JUl;R) • R ,

G - {e tY - exp ty | t * »} e R« .

Using the action of R* on R from Example 1, 3.6, we find the

curve

t » B (u) - (xetT,yetY) = u(t)e '

in R through u » (x,y) - The tangent vector to this curve at

any t is

Evaluation at t = 0 gives the map

Y* : E •* TK : u * (u,Yu) .

Evidently for this example Y * the zero vector at u = (0,0)

because all T * B have a fixed poiot there.

3.8 An action of a Lie group G on the right of a manifold P is called:

i) transitive if ( tfu.v « P) 3g ( G : E (u) = v ;

i i ) free if the only element of G with a fixed point is e ;

i l l ) effective if (H (u) « u , Vu * P) -» g = e .

-11-

"rrr-

3.9

1.3.9

2Example 1. The action of E* on P in 3.6 is not transitive, butthe right translation in any G is always transitive because:

u,v ( G -*• ( 3g = u-1v) r R (u) = v .g

Example 2. Let P = S » E* then the action induced by right trans-2

lation in K* is free, but the aforementioned action of R* on <R

is not free because every element of E* ha3 a fixed point at the

origin,

Given a Lie group G acting on the right of a manifold P , the orbit

of G through u * P is the set

[u] = {Hg{u) | g 6 G}

and we denote by P/G the set of all such [u] for u * P .

Evidently P/G gains a topology by requiring that the projection

Up : P + P/G : u * [u]

is continuous.

We shall denote the connected component of the identity in a Lie

group G by G . The notation is motivated by the special case

<J£(1;R) • R* and R* - GJt(l;(R)+ = {g t R | g > 0}

-12-

I.U.I

4. FIBRE BUNDLES

The general l inear group and i t s subgroups always have ac t ions on

tangent vectors on a manifold. The tangent vectors collectively form a

manifold, the tangent bundle, so ve have a ready-made action on this manifold.

However, at any point in a tangent bundle any element of the general linear

group simply sends the point to another one belonging to the same tangent

space, in a linear fashion. So the orbits of the general linear group in

this action are vector spaces. For wider application we shall be interested

in actions of Lie groups that give rise to orbits which are manifolds, and

not necessarily vector spaces. This is the motivation for defining fibre

bundles.

4.1 A principal fibre bundle over a manifold H is a manifold P with

a Lie group G such that:

i) G acts freely on P to the right ;

i i ) M = P/G and the canonical projection II :P •+ M is smooth ;

i i i ) P is locally t r iv ia l , that i s , every ifM has a neighbour-

hood U such that lC(U) is diffeomorphic to U * G .

We call G the structure group of the bundle ; property i i )

ensures that i t is transitive on al l fibres Hp(x) . We shall refer

to the principal fibre bundle as (P,G,M) or simply as P , when

G and H are well defined by the context-

Example 1. The t r iv ia l product bundle vith P = M x G .

Example 2. The frame bun rile or bundle of linear frames over a

smooth manifold M . Here we have

LM = ; (X±) an ordered basis for

G = GJt(n;R)> where n is the dimension of M . For all x f M

any choice of basis for T M induces an isomorphism t H - 8

which allows G£(n;R) to act \>y matrix multiplication on the

coordinate vector. Hence the subset determined by (x,(X,)) £ LM,

that is the orbit through this element (cf.3.9).is

How, as g runs through G4(n;ft) so E ( runs through all

bases for T M therefore ve may as well abbreviate [(x,(x.))]

to x . This leaves the required projection map in the form

-13-

I.U..2

: LM + M : U,U±))

Example 3- We' shall later have occasion to use LS . Here

S ana G = fi* and so LS = S x R* , the trivial productM = S ana G = fi* and so LS = S x R*

bundle. It has two components, corresponding to: positively

oriented bases L S = S x E , and negatively oriented bases- 1 ~ 1 -

L S = S x R . Quite generally, if M is an orientable

manifold then LM has two components; otherwise LM is connected.

h.2 Let (P,O,M) be a principal fibre bundle. For all y e $ the

fundamental vector field corresponding to y is y* , as defined

in 3.7. We observe the following properties.

i) Since the action of G translates each fibre along itself,

y is tangent to the fibre at all u*P . Also G acts freely

on P so y

the map

=* 0 t T P unless y = 0 • In consequence

has trivial kernel and since it is linear it is an injection

for all u sp {cf.lt.k below).

ii) Another formulation of y* is sometimes useful. For all

u * P define a map

0 : G •+ P : G * R (u)

u g

This induces a bundle map, Do : TG •+ TP , which at the identi ty

efG has the res t r ic t ion

*TG + T P : f * Y ,e u u

vhere we have used the isomorphism in 3.3 to identify the

s p a c e s TeG and "% .

i i i ) F o r a l l y s S s*1"1 " 1 1 u * P w i t h IL,(u) = x ,

This is another expression of the tangency of t to the fibr«

containing u , mentioned in i).

Example 1. flonsider the case of a frame bundle LM where

G = GJl(n;R) and <S » ^(n;R). Then we can display the above

maps in components as follows.

-14-

I.U

- TH(

(yj)* : IM -MPIM : u * (x^j.O.vVj1) .

Example 2 . Consider the ease of LS1 = S 1 x R* . We have the

following maps, for a l l y ( "%Jl(l,R) - fi and u - (x ,b )* LS

y* : IS1 * TLS1 : (x,b) •* (x,b,0,Yb)

: g *• (x,gb)

^LS1 : (1,Y) -+ (x,b,O,yb} , at g = 1 ,

and

a R :

It.3 Let (P,G,M) be a principle fibre bundle and let F be a manifold

on viiieh G acts on the left. Then the fibre bundle associated to

(P.G.M) with fibre F is a manifold (P x F)/G iefined by the

properties:

i) the right action of G on P x F is

: (u,a,g) "• (B (u), L (a)) ;

i i ) the projection map i s

fip : (P x F)/G * M : RG(u,a) * Hp(u)

and i t is reauirect to be smooth, with for any neighbourhoodU Of any j ( N

fi£ dlffeomorphic to U x F .

Eaample 1. Take F = G and use lef t tranGlation.

Example 2. The tangent bundle TM to a manifold M can be

considered as the bundle associated to the frame bundle 1*1 with

fibre fln , when H has dimension n . The fibre nj"(x) overL

each i 4 H can be identified with the tangent space T H to M-15-

We have

(LS1 x !R) LS1 x R : (x,b,a,g) H- {x.bg.g ""-a

for the right action of R* on LS x R . Therefore the orbit of

R* through (x,b,a)tLS * R is

[(x,b,a)] = {{x,bg,g~1a)[gHR*}

which we can identify with (x,a) * T 5 because as g runs throughx 1

all of R* so bg runs through all buses for T S

Example 3• The tangent bundle is a special case of the tensor

but

of its dual

F is a tensor product of k copies of R'n*

Here G

and h copies

GH(n;R) acts independently on the

factors of the tensor product, as for TM on fl and via trans--• n

We conventionally identify TIM withposed inverses on R

M x u, TtM with TM , and T°H with TM» the cotangent bundle.

When necessary we shall denote the projection maps onto M by

IL . Now we can formulate a tensor field of type (,) as a (smooth)v h ,

section of II . That is some w : M •*• TrM such that IL «w = I u .

The set of such form a vector space TrM for each k,h (cf.2.l).

Such amooth sections always exist for T?H, as witness the zero

tensor field of any type. However, the frame bundle LM need not2

have any smooth section, as witness the case M = S in consequence

of the Hairy Ball Theorem. In fact there does not even exist a

continuous section of T5 that is never zero.

Let (P,G,M) be a principal fibre bundle. For all u*P the vertical

subspace G of the tangent space T P is given ty the kernel of

{x

How we see that the map in k.2 i) actually induces an isomorphism

u ' 'u

so dim G • dim G .u

-16-

I.it.6

Example • For P = IS, , G = E* we have for all u = (x,t) t LS1

(x,b)

= E .

It .5 The canonical 1-form of a frame bundle LM is the map

9 : TLH + Rn : (u,X) •+• n -DlUu.X) ,"U. l i

where for a l l u = ( x . ^ H e IM

Hu : TM •* B n : a V - (&1) .

Since a l l vertical vectors l i e in the kernel of Dll_ , &{G ) = 0 forL u

al l u .

Example

© : TIS1 + E1 : (x.b.p.q) * p/b .

Or more generally, with matrix components0 : TIM* Rn : (x\bj,x\Bj) ~ (bj)"1 (X1) .

U.6 Universal covering manifold. Homotopic curves. Given a connected

manifold M , there is a unique universal covering manifold K .

iThat i s , a unique universal covering space {cf. 1.13) with manifold

structure.]

It turns out that (M,W1(M),M) is a principal fibre bundle over

M with structure group TT (M) , the f irst homotopy proup of M . I

For the proof see Steenrod [2U], pp.67-71, where the isomorphism

classes Implied by the uniqueness are elaborated also.

Example. Take Sn = {x t RQ+1 | ||x[ = 1} for n > 1 . Heal

projeetive n-spaoe, BF is the quotient of Sn by the group

0 * {ign.J} where J : x » -x .

n J™*' QIt turns out that S • RP ; the sphere is the universal covering

manifold of projeetive space.

We do not wish to develop details of the homotopy group here (cf.

Pontryagin [19], Chapter 9) . However, we do have occasion to use

-17-

1.5

in 5.T below the idea, of homotopy for closed curves. Essentially,

two curves c. , c are homotopic if one can be continuously deformedinto the other. More precisely, two closed curves c^ , c : [a,b]

i) = x are homotopic if there exists a continuous

map f : [a,b] x [o, l]- i( l such that

— H

with o1(a) =

f(a , t ) = f(b,t) = x , for a l l t i [0,1] ,

f(s,O) = C l(s) , f ( s , l ) = c2(») , for a l l s € [ a , b ] .

A curve is homotopic to zero if i t i3 homotopic to a constant curve.

5. COHHECTIOHS

The earliest formulation of a connection on a manifold was by Weyl in

synthesizing parallel transport of vectors along a curve. This was highly

motivated by applications to spacetimes in relativity theory where it is

essential to be able to compare vectors from different tangent spaces.

Connections give rise to geodesies, covariant derivatives and curvature and

in any particular ease there is no escape from calculating their manifestation

in coordinates, the Chriatoffel symbols. However, it suits our purpose to

adopt a more global characterisation to exploit the neat way that connections

partition the geometry of principal fibre bundles generated by Lie groups.

Expressions in local coordinates will come soon enough.

5.1 A connection V in a principal fibre bundle CP>G,M) is an assignment

of a subspace H of T P for all u * Pu

i) T P = H « G smoothly on TPu u u

s u c h t h a t :

( o f . U . 1 0 ;

i i ) DE (H ) = H,, , \ , f o r a l l g e G ( c f . 5 . g . Example 2 ) .g U H lUj

6

-18-

1.5.1

Thus V is a smooth distribution satisfying i) and i i ) . We call

H the horizontal subs-pace of T P or at u t P . I t turns out

that DIIp : TP •* TM induces an isomorphism H i T_ , >M for a l l

u*P . I t is common also to speak of V as a connection on M ,

i t being understood from the context which principle fibre bundle

V is in .

given by any Afc fl if ve put for a l l u = (x,b) t LS

^ . (of.[7])

The vertical subspaces E, , were given in k.k so for any

(x , t , p ,q ) t T, b}kS we have the decomposition

(x,b,p,-Abp) (x,b,O,q+Abp)

Evidently this i s a smooth decomposition on TLS

the right action of i t on LS i s

R : 1£X •* IE1 : (x.b) *+ (x,bg)6

Given any gf fi ,

so i t s differential is

DHg : TLS1 + TLS1 : (x.b.p.g.) ~ (x.bg.p.qg)

Therefore, as req.ui.red we have

DE (H, , .) - H, . .gv (x,b) (x,bg)

Finally, the differential of the projection JIT is

L

DnL : TLS1 •+ TS1 : (x,b,p,q.) •+ Cx.p) ,

so y i e ld ing the des i red isomorphismH (x ,b ) * T x S : Cx>b»P.-XtP^ ** <X>P>

Example 2. We observe that Example 1 is equally valid if S is

replaced throughout by B .

Note also that in both examples, H, .•, only looks horizontal

(in the standard embedding in F for LS1 or in K for LR) if

X = 0 . Finally, in the standard coordinate for either manifold, we

have the solitary, constant, Christoffel symbol T = A ,

-19-

1.5.2

5.2 Let (P,GSM) be a principal fibre bundle with connection V . Every

vector field w : M •+ TM has a unique horizontal l i l t wf : P + TP

such that for a l l u ( P

Dllp{wt(u)) = w-I and

Example. Take the case of a connection in a frame bundle LM . Then

V induces a derivation on vector fields on M . Local coordinates

about x«M induce a basis (3.) for T M for a l l y in some

Locally a vector field v : M •+ TM has an

The derivation induced by 7 i s locally givenneighbourhood of x

expansion v = v 3.

wnere u

r j i 8 k = ' a * 3 i ' • f o r a 1 1 t l J

are fields having components the Christoffel symbols. Now a field

LM •+ TLM : (x.b) * (x,b,X,B) turns out to be horiiontal i f and only

if in local coordinates

J[ef. 5.6,Example 2).

This says precisely that each member b^3fc of the frame b at

x«M satisfies the differential equation

where of course the component X in (x,b,X,B) is interpreted as

tangent vector and hence a derivation at x .

We can write the above symbolically in the form

B = X(b) - Vx(b)

Hence given any vector field vt TM with

w : M ->• TM : x -* (x ,X)

we obtain the horizontal lift wT t TLM with

-20-

1.5.1*

w1" ; LM + TLM : (x,b) ~ (x,b,X,XCb) - V (b)) .

This i s evident ly uniq.ue, and, s ince we have

DJIL : TLM + TM : (x,b,X,B) •+ (x,X) ,

the required projection property is achieved.

In particular for the constant connection X on S , we find

the horizontal lift

w*: LS1 + TLS1 : (x,b) -+ (x,b,X,-XbX) .

5.3 Horizontal l i f ts of vector fields have the following properties for

all v,w : M -» TM :

i) (v + vf = v f + wT ;

ii) for all smooth f : M + R define ff = f.IL, then f!v* = (f-v)* ;

iii) [v ,w ] = [v,v] ; (H denotes horizontal component)

iv) for all g*G and all u « P

vf(u) = vf(Rg(u)) ;

v) every horizontal vector fiela v • P •+ TPU is the horizontal

l i f t of some w : M + TM . *

WE are particularly interested in the following consequence,on curves.

5.^ Eveiy piecewise -C curve c : [0,1) •* M has a unique horizontal

l i f t c+ : [0,1) -* F such that : given

cf(0)

then

and the tangent vector field c is the horizontal l i f t of c .

Then the map

is a, dlffeomorphism for all t , called parallel transport along c

It commutes with the action of Q . •

-21-

1.5.5

Example• Again take the case of a connection V in a frame bundle

LH , developed in 5-2. We denote the restriction of V to a curve by

V (of.[6]). Then the required curve c has tangent vector field

c satisfying

= 0 .

In local coordinates this becomes

fe CcV • <=V ? rjk0 .

For a constant connection X = V * R on S consider the {very!)

typical curve with a« R

c : [0,1) • S1 : t •* at , so c(t) = « V t .

We have

cf : [0,1) + LS1 : t «• tat,b(t))

with the function b : [0,l) •+ R* satisfying

5.5

= 0at

Talcing the ini t ial condition b(0) = bQ we find

bCt) = bQ e"Xat .

I t follows that the parallel transport map is

n^(ot) : (0,bQ) H. (ot,b0 e"Xcft ) .

Thus for any a * 0 , while c proceeds round the circle S i ts

horizontal l i f t through {O.b.) spirals up or down the cylinder IS .

In fact the spiral remains in one component (cf. ^ . 1 , Example 3)

because the function b cannot take the value sero. Our argument

is again equally valid if S is replaced throughout by R .

A connection V ia a frame bundle IM determines geodesic curves

c : [0,1) •* M as the solutions of

Y.(fi) = 0

which in local coordinates becomes

-22-

5.6

1.5.6

It turns out that e is a geodesic if and only if it is the projection

of an integral curve of one of the standard horizontal vector fields

determined for all (X |(R n by (cf. Example 5.2)

LM + T L ^ : (x,b) «• (x,b,X,X(b) - \l*))

where X has components (X ) . •

The connection form of a connection V in a principal fibre bundle

(P.G.M) is the smooth map (cf. k.2, 5.1)

a i : T P - S : X H ® X ( J * Y , with y* = XG(u)

It has the following properties:

, <D(YU) " Y;i) for all

ii) u(X) - 0 •* X - Xgt TP ;

111) for all g « G and all vector fields X : P +

(il-DR (x) U)(X) (cf.

iv) connections and connection forms determine one another uniquely.

Example 1. Take the case of the constant connection * in LS

We know from 5.1. Example 1 that any X = (x,b,p,q) has the unique

decomposition

X - XJJ ® Xfi = (x,b,p,-Abp) » (x.b.O.q+Xhp) .

M o r e o v e r , f rom 3 • 7 and t h e e x a m p l e s i n h.2 we h a v e

Y*Cx,b) = ( x . b . O . Y b ) , f o r a l l Y * R •

Hence we r e q u i r e

wU/b.p. i l ) - Y - (q+Xbp)/T> ,

which is veil defined because b «1R* . The properties i)-iv) are

easily Been to hold for y • ^ e Just elaborate property iii).

1.5.6

Let geK* and X : LS1 •*• TIB1 : (x,b) -*• ( x , b , p , q ) . For a l l

g EG, Ad(g~ ) i s the i d e n t i t y map on f i e lds on (R*

Adlg"1) w(X) „-! V w(x)

Also

and

DH : TIS1 + TLS1 : Cx.U.p.q.) ^ (x .bg.p.og)

U(X) : B •+ IB* : a + (a,a.(q+Xp)) , by 3.3.

Therefore we find that the composite

ul-Dfi (X) : B* + TE»

satisfies as required

W=DR (x.b.p.q.) » O)Cx,bg,p,q.g)

= (qg + Xbgp)/bg

Example 2. We develop further Example 1 in k.2. From 5.2 we find

expressions in local coordinates as follows. Let u = (x ,t ) < LM .i i _i i

Then a typical tangent vector at u appears as Y • (x ,b,, x ,B.)

and is partitioned by a connection with components T into the

direct sum

Nov, in components, <JJ(Y) * ii.(n;R) is that matrix ty±) - y such that

CYi) - - < From Example 1, U.2 we find that this requires

Or in matrix form

Y = (B +

Next we observe that for all (g.) = g «GJl(n;R)J

Therefore H)(DE (Y)) i s t h a t CnJ " n :

-23- -2k-

1.5.7

Or in matrix form

,-1n = {gB + gbrx) (gb)"

= g(B + bFX) b^g"1

which by the linearity of the action of g gives as required

= H = AdCg"1)O

In passing note that the expression for DR (Y) demonstratesg

property 5-l.ii)•

5.7 let (P,G,M) be a principal ffbre bundle with connection V . We use

the term "curve" to mean a pieeewise continuously differentiable one.

The loop space at any x t M is denoted C(x) and consists of all

closed curves starting and ending at x . There is a natural product

on such curves. For all c < C(x) we have by 5-3 the parallel

transport isomorphism

and it commutes with the action of G on the fit re. The set of

all such { T |C « C ( X ) } , forms a group 4(x), the holonomy group of

V vith reference point x . We can realize *(x) as a subgroup

$(u) Of G for any u ( P , as

*(u) = {g4(j|R_{u) = I (u), TB C c

Equivalently, if « is the equivalence relation "can be Joined by

a. horizontal curve" on P , therefore

1.5.7

Example 1. Consider 1M with constant connection A . From 5 .U

we observe t h a t for a l l j i E , i f e i s a closed curve in C(x) then

T i s the i d e n t i t y on lC(x) • How K* ac t s freely on the frame

bundle UR and so for a l l (x,b)£ Iff!

4 U , b ) = {8* K*|H Cu) = u} = {1}

tfe know t h a t ho r i zon ta l curves in LR are of the form

c f : t

hence (x,b) ~ ( x ' , b ' ) i f and only i f for some a t = r* R

+ a t , be"Xat)

x ' - x = r and b ' / b = e"

Example 2. Here we see a departure of the geometry of from that

of IB, in the form of a n o n - t r i v i a l holonomy group. Again use the

constant connection X . We take S t o be H modulo the i n t e g e r s ,

2 . We s t i l l have t r i v i a l members of C(x) , for any x * S , but now

we also have those curves which may make one or more c i r c u i t s of S ,

l i k e

c, : [0,k] •+ S 1 : t •* (x+t) (mod l ) , k* Z

In fact the essential members of 4(x) are in the set

•ET |kea}

and we know that these elements commute with all E because

g

Tc : n*(x) - j£U) : (x,b) » (x,be"Xk) .

Accordingly the holonomy group with reference point (x,b) is

t(u) = {g( 0|u — R (u)}

Then i t follows that for a l l

, *(Hg(u)) =

i i ) u ~

iii) $(u) is a Lie group with identity component $ (u) where

* (u) = R (u) ; c is homotopic to zero in C(x)}

is called the restricted holonomy group of V vith reference point u. •

-25-

<S(x,b) = {e~Xkt R*|k* Z} .

The relation ~ has the same form as in Example 1 but on IB ve

have (x'-x} (mod l) = 0 whenever (x'-x)t 2 . Therefore in a l l

fibres n!~(x} we findJj

(x,b) - (i,be~Xk) , k*Z .

-26-

1.5.9

We knew from 3.1* that for a l l g~ i R* , ad(g ) is the identity-map on E* and so

Also, as required

t(H (x,b)) = »(x,bg) = t(x,b) .

The identity component is t r iv i a l

*°(x,b) = { e " ^ R*|k = 0} = {1} .See the Example in II.3.12 which has holonomy group «• .

5.8 Let (P,G,M) be a principal fibre bundle with connection V . For a l lu*P the holonomy bundle through u is the subbundle

P(u) = { v i P J u ~ v }

with structure group *(u) . Since — is an equivalence relationthe holonomy bundles parti t ion P into non-empty disjoint se ts .Moreover, every g t G maps each horizontal curve into a horizontalcurve; we have the isomorphisms

Rg : P(u) + P(ug)

adCg"1) : *(u) •*

Example. For the preceeding Example 2 we find

F(x,b) » {(x+r, be"X r) | r« E}

and i t s structure group i s 3 .

5.9 The Levi-Civita connection of a Riemannian or pseudo-Riemannian manifold,H with metric tensor field B*TpM, is the unique connectionin LM such that :

i ) parallel transport i s always an isometry along curves in M ;

i i ) for a l l vector field* v.w : M - TM

vyw - vffv = [v,w] . • (of.5.2)

The f i rs t condition is referred to as compatibility with g , thesecond is referred to as the symmetry or torsion-free property of theconnection.

-2T-

1.5.10

Example 1. The constant connection X i n LR is t h e Levi-Civita"

connection induced by the Riemannian metric tensor field given2Xx

at x« R , that islocally by g = e

H : ((x,y) ,(x,z

However, the constant connection X in LS does not arise as the

Levi-Civita connection of any Riematinian metric on S

Example 2. Consider the cylinder, useful in the sequel, given by

N = {(1)1,0) « E2|IJJ * (0,SIT), tJ t [0,2TT)} - (0,2rr) x s 1 ,

where we have identified a = 0 and a = 2TT . So S appears asthe real numbers modulo 2ir , We can supply this with the pseudo-Riemannian metric tensor field g with coordinates in the aboveindicated chart given by the matrix (cf. [1] and [5])

(g j j ) = U-COSl(j) , at t H

From the well-known local coordinate form of the above theorem, the

components of the induced Levi-Civita connection are given by

" hAnd so we find at

, r l ) sin* I1 °u i j ' " 1 - COB* [0 1

, .2 , _ i i n ! - 1° X

* v i j ' 1 - cos* [1 0

5.10 The bundle of orthonormal frames of a Hiemannian or pswudo-Riemannian

manifold (M,g) is the subbundle 0M of LM , consisting of ortho-normal

frames

OK 11*1 ^ ,X <5±J}

Hence a connection in LM induces a connection in CM . •

Example• Consider the pseudo-Riemannian cylinder (S,g) in thepreceeding example. The component OH of positively orientedorthonormal bases has the structure group mentioned in 3 . 1 . , Example 2,

We readily observe that- 2 8 -

1.5.10

1 - eosiji

is an orthonormal basis for T

this at x " 0 (cf. 3.1) then

coshx sinhx

, . iH We can arrange to locate

sinhx coshX 1 ~ e o a *

Hence we can think of this bundle as

0+S = j (i|/,0 ,x) ' (0 ,2TT) x S1 x fi = H x S .

We see that X£ 8 determines at (i|i,ff) t N a basis {•A'd ,-Ai ) ,

which we have expressed with respect to the basis (3.,9 ) induced

by the coordinates (IJJ.O), such that

1 - COSljJf. [5])

I t follows from 5-1* tha t any curve ? [0,1] * N has a unique

hor i zon ta l l i f t t o O N . I h i s ia given by

c* : [0 ,1] + 0+N : t •* ( c ( t ) , x ( t ) ) ,

where the r e a l functioii "X s a t i s f i e s

dt= _ c

2c

sin c

1 - cos c

which equation is implied by

V*iV = ° ' i

or equivalently

It 4 - -rL'*£ • i

vhere (the function) x determines the

-29-

1,2

before.

1.5.13

We find tha t , for example,

c : t >-• (4io,t) , IJJQ = constant

has horizontal Lift through x(°) • XQ a t t = 0 giveo by

c f : t t-jdlig.t.XQ-toig), a0

The corresponding parallel transport is essentially therefore

sin*.

•*0 *0 ~ " 1 - cos*Q

which is evidently an isometry for g by construction, and in

consequence of the identity

2 2cosh x - 3in X = !

5.11 In a Eiemannian or pseudo-Hiemannian manifold (M,g) , for a l l xf M

and a l l v «T M there is a unique geodesic curve c in M such

that e{0) = x , c{0) = v; (cf. 5-5)-

The map exp is defined on the subset

E » {vtTMsli 3 geodesic c : [0,1] + H with C T (0) - x, cv(0) - v}

by putting

It is always smooth on some neighbourhood of 0 *T H . •

5.12 A connected Riemannian manifold (H,g) is complete (cf. 2.h) if and

only if it is geodesically complete with respect to the induced Levi-

Civita connection. By geodeaicolly complete we mean that every

geodesic admits an extension to arbitrarily large parameter values.

A connected complete Riemannian manifold has the following properties:

i) For all x* M> the map exp : T M — * M is surjective;

ii) Any two points can be joined by a minimizing geodesic. •

5.13 A connected Eiemannian manifold (M,g) is complete if any of the

following conditions hold:

i) All geodesies starting from any particular point are complete;

ii) M is compact;

-30-

5.1k

1.5.Ht

iii) The group of Isometriea is transitive on M (cf. 3.8);

iv) The orbit of the group of isometries through any particular

point in H contains an open set of M . (Then the open set

coincides with M and ve have iii). m

^ ° e cuntature of a connection V in a frame bundle LM is the map

on pairs of vector fields defined by

H : : <v,w)

ft, . :(v,w) * ~ V V > - V V v z ) - VEv,w]*

I t follows that we can consider the curvature as a tensor field of type

U , so R'T^M . If V is the Levi-Civita connection of some

g tTJl then we call R the Biemann tensor.. The curvature has

a variety of well-known geometrical attributes and Riemann tensors

in particular possess a range of symmetry properties, we shall recall

bhese as necessary in the sequel.

The Ricci tensor is the unique (up to sign) contraction of the

Rieraann tensor to a symmetric member of T M . In the theory of

relativity i t is used to link the physical energy-momentum tensor to

the geometry of spaeetime (of. Chapter I I I ) . We fix the choice of

signs by indicating local components:

components: R*kJ = 3 ^ - 3 ^ ^ + 4 j " 'jm ik '

Ricci components : R.

Scalar curvature : g

-31-

I I .

II

CCMECTIOH GEOMETRY FROM FBAME BUHDLES

We have seen how a connection in a frame bundle LM decomposes the

tangent spaces into a d i rec t sum. The hor izonta l component apana 'd i rect ions

in the manifold M" and the v e r t i c a l component spans "directions in the fibre*

tha t i s in &f(n;R) . We reca l led t h a t a Eiemannian or pseudo-Hiemannian

metric tensor f ie ld on M always determines the Levi-Civita connection in LM.

In a spaeetime manifold M we have, impl i c i t ly determined from the d ispos i t ion

of mat ter , a Lorentz metric tensor f ie ld and hence a connection. Also, when

i t i s convenient, we can work on a spaeetime with a sub-bundle of LM , namely

the "bundle OM of orthonormal frames with s t ruc ture group the Lorentz group.

Sometimes, to reduce the dimensions s t i l l fur ther , we can deal with submanifolda

and t h e i r bundles to gain useful information about the whole spaeetime.

In order not to confuse the geometry t ha t a r i ses from a connection with

tha t which requires a metric tensor f ie ld on the underlying manifold, we avoid

spec ia l i s ing to spacetimes where appropriate . Thus we consider a manifold M

with a connection V . This V induces a Riemannian metric on the frame

bundle LM in such a way as t o make hor izonta l and v e r t i c a l subspaces or tho-

gonal. We are here in t e re s t ed in the consequent geometry of LM and the

Cauehy completion of i t s connected components. The precise def in i t ion of a

spacetime i s given in Chapter I I I where we consider the geometry induced by i t s

Levi-Civita connection.

1. SCHMIDT'S BUTOLE METRIC

We suppose t ha t M i s a smooth n-manifold with a connection V in LM,

The ideas o r ig ina te in Schaidt [21] though Hawking [lk] has suggested an

e a r l i e r or ig in in Ehresmann [ l l ] . However, when asked by the present author,

Schjaidt was unaware of any influence of t h i s work of Ehresmann on h is

construct ion.

1.1 Existence. We denote the standard inner product on n for any m

by • and we r e c a l l the canonical 1-form Q from k.5 of Chapter I

and the connection form Ul from 5-6 of Chapter I . Then the

Riemannian metric defined by Schmidt on LM i s denoted by < , >

where

TLM x TIM •+ ft : (x,Y) »• + u(X)'U>(Y) .

Symmetry and b i l i n e a r i t y follow from tha t of • and the l i n e a r i t y

of ® and a) . Posi t ive def ini teness follows from tha t for •

because for a l l X * T LMu

-32-

II.1.1

|x|| • <X,X> = 0 -» ®(X) = 0 ( Rn and

-»X £ Q and X t H (by I . I t . 4 . 5 . 6 )u u J

* X - 0 . «

Example, Take H - S vltli the constant connection \ . From I.1*.5,

5.6 for a l l U.b.p.q.), U,b,r,s} t T ^ ^ j l S 1

< Cx,b,p,(i),U,-b,r,s)> = tpr/b2) + (q + Xbp)(9 + Abr)/b

| U , n , P . 4 ) | 2 = (pA>)2 + « g + XUpJ/b)2 , (cf. [ 7 ] ) .

We find the length with respect t o the above norm of two curves in

LV .

Case 1. c : t •* (xQ,t) , for fixed xQ « S1 ana t « [l , tm] .

This is a vertical curve vith tangent vector

cCt) - Uo . t ,O,l) ! l |c(t)| = 1/t .

Hence the length is {for t > l )

j W JcCt)|dt = log

1

Since t h i s tends t o i n f i n i t y as t •* » we see tha t a l l of the f ibresm _

+ t ^ ) S L S 1 are infinite for xQ t S1 .Case a. e : t •* (- t mod 1 , e ) . f o r t « [0 , t ] .- —^^— m

Again t h i s begins a t bas i s 1, but t h i s curve by 1.5.5 ia hor izonta l

and for t > 1 i t necessar i ly meets a l l f i b r e s . We find the tangent

vector

c ( t ) - (-t mod 1, e X t , - 1 , XeXt)

so-Xt

( 1 -

Plainly this tends to 1/|X| as t •* •» . So ve have an example of

Hence the length is (for t S- 0)

II.1.2

a curve of finite length in L 3 whose projection covers 5 in-finitely many times.

1.2 Uniqueness. We prove the claim of Schmidt tliat if • is replaced by

different inner products on Hn , Rn then tlie ensuing metric struc-

ture for L'M is uniformly equivalent to that given 'by <. , > above.

(Cf. 1.2.1*, i . i i . )

Proof. Consider the videst generalization of < , > . It will be of

the form

« , » : tX.Y) ~9(X) • ©CY) + u{X) * «(Y) ,

where • , • are inner products given by

2 g* : ^ x R11 + K : CCu^.Cv1)) * G u V

•^ J

for sone matrices of fixed numbers (g.,,). (a,,) •

Let d and d' be the topological metrics (ct.l.2.h) induced by

< »> > <f i* respectively on each connected component L'M . We

must prove that for all u,v * L'M

i) (^e1 > 0)(3S1 > 0) : dtu.v) < fix-» d'(u.v) < ex ;

ii) ( V E 2 > 0)(352 > 0) : d'(u.v) < 52^, d(u,v) < E,, .

From 1.2.1* i t is sufficient to consider the norms I I and I I ,

induced by { ,> and «,» reapectively on an arbitrary tangent space

TuL'M . Suppose X i TuL'M and ©(X) = (h1) f ft" , U(X) = (v1) t fP .

Then ve have

New (gi ) determines an element of GH(n;R) which sends (h ) to

(6jin ' * Rtl a n d i n ' t h e standard norm | | for fP we have

where m • det{g. ) . We find a corresponding m_ and aimilar

equations for (0.,} . We knov m and IU are positive by de-IJ g *

finition of the inner products. Also, for some direction cosinesa , ou < [0,1] not both zero if X is non-zero, by positivedefiniteness,

- 3 U -

II.1.3

m a I6. B1

We suppose first that m a 5 m_ct-g g (J G

froml-u-LI ^ Bird ^

Then the required result follows

For,

i ) |x|— < r whenever

that la whenever |x|| < r

ii) < r2 whenever )x||»/mgag <

that is whenever

g a g

< m a r .

Plainly the case ma 4 m.a_ can be handled also. •g g Q G

Example. For S with the constant connection \ we have for anym -BL, > 0 , a = ou = 1 and so

<<(x,b,p,q.),U,b,r,s)» = m pr/b2 + m^q + Xbp)(s + Abr)/b2 .

1.3 Uniform action of Gf(n;R). For a l l g t G , the identity component

of G((n;R) , the right action H is unifonnly continuous on themetric space L'M .

Proof. Results above allow us to give a brief complete proof differ-ing somewhat from that outlined by Schmidt [21]. Firs t ly , we observethat by 1.2.1* the metric topology coincides with the manifold topologyso the proposition is well-formed.

It i s necessary to prove (of. 1.1.9) for a l l g t G and a l lu,v < L'M that

(Ve > 0)(3fi > 0) : d(u,v) < 4 =•» d(R (u),H (v)) < E .ft •&

From I.2.U i t i s sufficient to show for a l l Y £ T L'M and al lu

u e L'M that

0 ) ( 3 6 > 0) : |DR

We follow Example 2 in 1.5.6 and write Y locally in matrix components

aa Y = (x,b,X,B) , so DH {x,gb,X,gB} , From I.I*.5 and 1.5.6

-35-

Since

above

9(Y) =

m(Y) =

( D E ( Y ) ) =

det(g)"1

gives

b^x , 6

(B + bFX)b

g(B + bPX)

= ctettg'1)

I I ,

g

- 1

> 0

,1.3

(Y)) =

i

- 1 .

, th

|DR ( Y ) Is

where again | |case that det(g ) f 1 then)

standard norm on R aisd E . In the

Otherwise det(g ) > 1 , vhen we have

|DRg(Y)|[2«: dettg"1) |Y|2 .

—1 1/2So for a l l e > 0 choose 6 = e / ( l + det(g )) whereupon

|DHg{Y)l|

Therefore DB and E are uniformly continuous for all g « G . I

Corollary. Uniform extension to completion.

From I.1.10 every B has a uniformly continuous extension B

to the Cauchy completion (L'M,d) in which the metric space (L1H,d)

is dense, cf. 1.2,1*.

The extension d ia also well defined and unique by the unique-ness (cf. I.l.U) of limits in a metric space, which ia of course alwaysHauadorff. For similar reasons B is uniquely determined becausefor a l l u ( L'M there is a unique l imit ,

lim R (u) , in L'M .6

From I.1.10 we recall that B agrees with H on L'M ,

-36-

II.1.3

It follcv3 that for all u,v £ L'M and all g t G we have the

bound

0. d(u,v)

Example. From the Example in 1.1 we see that for any g e R

(p/b)£

(q

LSThe geometry of LR and LS with the metric tensor field < , >

induced by a constant connection X € R has been described by Dodson

and Sulley [7] . I t turns out that for X ? 0 the essential part of

Ifi is uniformly equivalent to

with the standard metric, under the map

f : LR * R2 : {x,b) * (Xx + log |b | , ( l /b + sgn(b) )/|X| )) ,

Horizontal curves appear in ^(IH) as the lines u = constant end thefibre II (x) is the curve given by

For any g « R* the right-action appears in f (LR) as

^•E^f" 1 ; (u,v) H- (u + log|g | , (v/g) + sgn(v)Ugn(g} -

The length of the horizontal curve through x - 0 , b = b is l/ |Xb| ,

Evidently, Cauchy sequences in <j?(LR) with Uni ts on the lines

v » ± l / | 11 establish the l a t t e r as the 'boundaries of the two components

of IJR . The extension of E to LB is then given Toy the action on

V * (u + log|gj,±sgn(g)/|X|) .

We can use <p to define a uniform equivalence of the essential

part of LS1 with the cylinder obtained by identifying points (u,v)

and (u + Xt,v) , k € Z . flere the boundaries of l/V1 and ITS1

appear as the circles v • ±1/|X| . Plainly the above action of

R and i ts extension also applies to LS by taking u modulo X .

- 3 7 -

l.h

Il.l.h

-± -± 1For L E and L S the lioundary is itself an orhit of the identitycomponent R oj 1* ,

The b-boundary. Oxur preceding result and corollary show that the

identity component G of G£(n;R) acts on the right (cf.I.3.6) of

each manifold L'M . Hence from 1,3.9 the topological space

H = JJw/G+ is well defined, with

I t , : L'M + M : u •+ {E (u)|g t G+}

continuous, and coincident with IL, on' L'M . Thua we have

11 , (L'M) = M and we define the b-boundary of M , with the given con-

nection, to be 3H = M •» M . From the previous results of this

section 3H may be non-empty and i f so then i t is essentially unique.

We shall call M the bundle-completion of M , with given

connection.

Example 1. For R with constant connection . we have from above

(Example 1.3)

L+R * Uu.v) € R 2 | v

The orbits of g e R appear as the exponential curves with

one curve corresponds to each fibre II (x) . For each x « R suchL

a curve approaches the boundary v = 1/|A| only as u -* -•«• . Also,an open set consisting of such curves and containing the boundaryv = 1/]A| must include a l l curves for x in an interval (-™,a) forsome a t E . Hence R = L B/R is homeomorphic to a halfVclosedinterval.Example 2. For S with constant connection X we find

The orbits of g i K appear as exponential spirals on th is half-

closed cylinder and again the b-boundary consists of just one point.

However, unlike the case in Example 1, any neighbourhood of v = 1/JX]

in L S intersects every fibre. Eo the only neighbourhood of tile

b-boundary point of S i3 S , Hence S is at most a T -space

and therefore not Hausdorff Ccf. 1.1.1.). We can see that , in u-v

coordinates with the standard metric on the cylinder,

n * ( X a . t e " ^ + l ) / | X J )

- 3 6 -

II.2.2

is a Cauchy sequence in the above isomorph of L S

obviously Cauchy in the original x-b coordinates:

n *• Ca.e ) ; cf. 2.5 below.

But it is less

II.2.It

this is a union of open sets and therefore open,

Suppose a t R (U) ; then a-R (b) for some b< U . HenceS fi

2. THE METRIC TOPOLOGY OF L'M

As a metric space L'M is necessarily normal (cf. 1.1.7) and therefore

Hausdorff. By construction L'M is connected,locally connected and arewise

connected. It also satisfies the second axiom of countability: the metric

topology on L'M has a countable base. Here we shall collect some further

consequences of over choice of metric. Most important is that the holonomy

bundle when factored by the holonomy group also yields M ; this is due to

Schmidt [21] though our approach to it is a little different from his. We

shall make occasional use here and subsequently of the survey article on fibre

bundles by Eells [10]. At the same time we should mention two standard texts

that are invaluable reference works: Fontryagin [19] and Steenrod [24 ]•

2.1 Q acts transitively on fibres

Proof. Firstly, G acts freely on L'M by construction as a principal

fibre bundle and transitively on L'M because L'M consists of ordered

bases. (Cf. I.3.S, U.l.) Now suppose that x t 3M so

II—, (x) €. L'M •>• L'M . Hence we may suppose that there exists some

UQ € It, (£) , Then from 1.1* above

n£.(x) - {Rg(u0)|g < G+} .

If u,v f IU,(x) then we have for some g,h fe G

Therefore

S.2 TI-. ia an open map

Proof. (Cf. Bells [10]) Suppose \J is open in L'M ; we show that

IIs-, (U) is open in M , Any g £ G is a homeomorphism so:

B (U) open and hence 5 ( 0 ) is open.

From 1,1* 11 , (U) is open if and only if 1^,(11^,(0)) is open in

VM . We show that

-39-

2.3

2.1*

Conversely, suppose a. f n=t(n=,(U)) . Then TI>,{a) =IT7,(b) for

some b e U . But from 2.1 G acts transitively on fibres 3o

R (b) = a for some g t G . Hence a t R CO) . •£ S

Completeness of fibres with the induced metric. Suppose that (v )

is a Cauchy sequence in L'M with v t nL,(x) for all but finitely

many n t S .

Then there exists v = lim v £ lit, Ox) .

n L

Proof, fly construction L'M is complete so v 4 IU,(y) for some

y t M . Suppose y 5* x . Distinct fibres are disjoint, so for some

r > 0 , denoting the extension of the metric to L'M by d ,

Then the open ball S{v,r/2) contains v but does not meet IL,(x)

Hence (v ) does not meet thi3 bal l . That contradicts i t3 con-n

vergence to v so x = y . •

Corollary• If (v ) is a Cauchy sequence in L'M , contained wholly

in H (x) then (v ) converges to someL n +

by the transitivity of G on fibres,

in this fibre. Moreover,

(Vn t H)(3gne G+) : R (vj = v .

Hence, by the continuity of each R

lim e , the identity In

We see another aspect of the completeness of fibres over M in

the next result. Note that here we are measuring distances by taking

infima over curves in the fibre for the Cauchy condition, in contrast

to the situation in 3.h below.

Fibres in L'M are homogeneous spaces. Every fibre IL.t(x) ^3 a

complete Riemannian submanifold.

Proof. The metric tensor field < , > restricts to JIT, (x) which is

-1+0-

i,.'1.— • '.hi "A !. 1

I I .2 .5

therefore a Eiemanniaii suhmanifold of L'M . Let u t L'H . Now

i t I d CiCx) then T i s vertical and so, by definition 1.1,

(JJCY)|( , in the notation of 1.2. I t follows from the proofin 1.3 that for a l l g 4 G

II.2.5

c^ : [0,1) - L'M : t * (c(t) .^0^)1 , With cttQ) - ^ = U±,\) •

Mow we have a sequence U Ct )) in L'M with projection

(x ) = (1IT . tv )] in M . The action of G is by £.1 transitive onn -Li n

fibres so for a l l n t II ve can find

Therefore the set of maps

(DB | g £ G+}

gives rise to a group of isometries on T II (x) . Since G acts

transitively on such fibres, by 2.1, the fibres are homogeneous. It

is known that homogeneous Riemannian manifolds ore complete, see for

example Kobayashi and Noaimu [lB], p.176. (Cf. I.S.li.) •

Example. Consider L s with the constant connection X . Then a

typical tangent vector to the fibre over (x,b) £ L E Is Y = (x,b,O,q).

From our previous results we have for any g t ft , DR ( Y) = (x,bg,O,qg)

and sok | / h , |DH (Y)J = |qg|/bg =

D

For any x e S the fibre HL+(x) is complete, with infinitelength, in the induced metric {of. 1.1). However, II +(x) is in-complete and finite when considered with the full metric which (cf.3.*0 allows points to be linked by horizontal curves that may leave

2.5 Boundary points generated by horizontal Cauchy sequences. Every xin the b-boundary 3M is the projection by Ilr, of an equivalence

class of Cauchy sequences on L'M . If the Cauchy sequence (v )

determines x € 3M then th i s x i s equivalently determined by a

Cauchy sequence (u ) on a horizontal curve in L'M .

Proof. This result was obtained by Schmidt [21] through a fibre iso-

metry with L'lr which is a complete Riemannian manifold. We give

a direct proof of similar length.

„-; There la a curve k : [0, l) •* L'M such that , for some sequence

(t ) c [0,1) convergent to 1 , v = t(t ) . Sow k(t) does notn n _ n

eventually lie vholly in- one fibre H ,(z) for then by 2.3 (vQ)

would have a limit there instead of in the boundary L'M ^ L'M ,

Hence \,'k - e : [0,1) • M is inextensible in M . We take the

unique horizontal lift {cf. 1.5.It)

From the property 1.5.1 i i ) of a connection we know also that elements

of G will map a horizontal curve into another horizontal curve.+ n2

G has the standard topology induced by His the limit of (v ) ; i t is known to be InSuppose that

L'M "» L'H . By completeness of L'M the curve c has an endpointu^ £ VK "* L'M with

nE,Cva>) = xnE,Cva>

Hence there exists some g^ £ G with H (u^) = vM . By con-

tinuity the sequence ((L,) defined above converges to g_. Let c*be the unique horizontal lift of c through B (v^) , that is

c*(0) = R (v. ) . Recall that R coincides with R on L'M .

We define the sequence (c*(t )) = (u ) on the horizontal curve

c* . Now, parallel transport commutes with the action of G

(cf. I.5.1*) and 30 e* = R » e* , Therefore for all n € H ,

u = H (cT(t )) . We find that (u ) is Cauchy with the same limitn _gj, n n

v^ t L'M •* L'M as (v ) , because:

a(R H • d(RR R ct(tk)).

Mow, (v ) is Cauchy, and ( g ) c G is a sequence of continuous maps

with limit

that

So for all e > 0 we can find N£« H such

for

Example. Consider L 5 with constant connection X / 0 . We have

L S = {{x,b) e S x R } and a Cauchy sequence in L S is given

) for some fixed a e S .for example by (u ) : a •* (a,e )n + 1

Plainly this has no limit in L Sn,k t H the points u ,

n

for some fixed a e

I t is Cauchy because for anyl ie on the curve

-kl- -1*2-

II.2.6 II.2.6

(L'M, < , y ) . From 2.5, for all 5 «SM we can find a Cauchy

sequence (u. ) on a horizontal curve in L'M withn

lint (11 ) = x .n

So we can take the Cauchy ccagpletion L'M(uQ) and extend the action

of ^{UQ) uniformly continuously as before and find

c : t ** (a - t,iXt,

L'Mtu0V*Cu0) M = MU3M .

Remarlc. Ihrig [16] has found the holonomy groups of a large class of

spacetimes.

Example. L S vith constant connection X , again! (Cf. Example

2 of 1.5.7.) Let UQ » (XQ.^Q) *• L^S1 be arbitrary, then we have

- { e - ^ t 8 + | i « 2 ) ,

- {(xCmoa D . b ^ ^ " ^ ' ) | , ( B } .

Front Example 1.3 we have the isometric equivalence

^ - ^ / b j A l ) + 1/|X|) | x i R} .

This allows us more easily to see the completion

L V d i g ) 5 {(Uxjj + log b0)modX,v) | v > l / |X|} .

The action of *(uQ) was given in^ilieitly in 1.3; i t easily extends to

5 -Ax((XxO + l o g

log b0)modX,(v -

Hence we find the quotient (cf. 1.3-9) from

Xk+ log DQ)modX,v)] = {{(XXQ + log b0)modX,(v - 1/|X|)e*" +

log

As required this agrees vith (cf. Example 2 of I.1

L V V R + a s^-U {£1} .

with tangent vector field

c : tn - (-l,XeXt) and Je( t ) | - e~Xt , by 1.1,

therefore by 1.2.4,

fk -Xt ,. •• 1 I -Xk -Xn |j . d t - _ | . - . | .

In fact by 1.5.4, the curve c is a horizontal curve: the horizontall i f t of R - » - S 1 : t * * a - t through 1 € fi» at t = 0 .

Note that in the uniformly equivalent netric apace

{(u,v) « S1 x (1/ |X| , -)} a L V

our horizontal curve appears as (cf. Example 2. in 1.4)

u(t) = constant = X(a - t) + log e

v(t) - (e"Xt +

Xt Xa

Also, our Cauchy sequence becomes

and we can easily see that i t has the limit

with respect to the usual metric.

2.6 Holonomy bundles generate the same b-bound&ry. The holonomy bundlethrough any point in the frame bundle determines the same b-boundarya3 the frame bundle i t se l f .

Proof. Schmidt [20] pointed out the equivalence of this result with

our 2 .5 .

Let u fe L'M be arbitrary and denote the holonomy bundle throughuQ (cf. 1.5.8) by

L'M(uQ) B {T t L'M | UJJ i- v}

It inherits a metric from inclusion in the Riemannian manifold

-43-

II.3.1

For, tlie point JJ corresponds to the singleton class

log T

and as v runa through the set (l/[X|,™) so the corresponding point

(x,b) i 1 S runs through values given by

Ax + log h = Ax. + log b

b = l/(v|A| - 1) , cf. Example 1.3.

Hence x runs through ' {Ax + log t>0 + log(v|X| - l ) | v > 1/ jX| > ,

which means that modulo 1 i t certainly completes a circuit of S .

3 . THE QUOTIENT TOPOLOGY OF M

For each connected component L'M and i t s Cauchy completion L'M we

have induced e s s e n t i a l l y t he same quot ient topology for M = L'M/G . In

fact H has a sennbnetric topology contained in the quot ient topology. By

cons t ruc t ion H i s connected, l o c a l l y connected and s rcv ise connected. I t

i s second countable but need not tie l o c a l l y compact nor more than TQ . In

t h e case t h a t M1 i s a Riemannian manifold then the bundle completion H v ia

the Levi -Civi ta connection coincides wi th i the d i r ec t Cauchy completion; t h i s in

p a r t i c u l a r persuades us of the mathematical s ign i f icance of the b-boundary.

3 .1 The aemlnetrlc structure, p . There is a semimetric p for M with

p(x,y) - inf{d(a,b) £,(x) , b * H±, (y)}

and the p-topQlogy ia contained in the quotient topology.

Proof. Plainly p is a semimetric. It determines a topology from

the open balls of radius r , centre x t H given by

SCx.r) * {y fe M | p(x,y) < r} .

The quotient topologfr is the family of sets

{UfiM | n£,{0) i s open in VM} .

We show that the quotient topology contains the p-topology by proving

that I t , is continuous in the p-topology. Without loss of general-

ity consider an arbitrary x t M and an open set A of the form

-45-

3.2

II.3.2

A = fflf1(b)£ M | d(b,nt, (x)) < t }

= {y f M | p { y , x ) < e } , f o r some £ > 0 .

We f i n d an open b a l l B { a , r ) a r o u n d any a f l j A ) w i t h

- <- L X

d(a,HT,(x)) = s , Since s < £ . we can find a suitable r > 0 with

r < e - 3 . Therefore

a£ B(a,r) = {c <= VM \ d(c,a) < r} t l l j , ^ ) ,

because

d(c,Hj, (x)) ^ d(c,a) + d(a,I^t{x)) ,< r + s < e . m

If L'M ia complete and p is a metric then M is complete

Proof. (Cf. Eells [10], p.63.)

Suppose that L'M i3 complete and let (x ) be a Cauchy sequence In

M = L'M/G+ .

Using a subsequence if necessary, we suppose that

p < v W *C l / 2 ) n •Choose any b 1 t II t (x1) and b 2 1 n^, (xg) such that d(bisb2) < 1/2.

This is possible because ty • we know that

Choose

sequen

a limit there,

1/2 ,

such that d(b2>b ) < (l /2) and so on. The

sequence (b ) so formed is Cauchy in L'M and go by hypothesis has

Define

b^ , which ia unique.

= 11 , (b^) t M . Then we have

Hence (x ) converges to x . •

Remark. We know from 1.3 that G is not an lsometry group for L'M ,

though it does yield isometries on the individual fibres by 2.k. An

isometric action would have caused the semlmetric topology to coincide

with the quotient topology (ef. Eells [10], p.62), However, the

action of G is uniform and in consequence It may be possible to

•xploit the isometric action on vertically separated points, in

|rtudi«e of bunches of fibres over appropriately "small" sets in M .

-U6-

3.3

3.i

II.3.U

M Is not Tj if the orbits of G are not closed in L'M

Proof. We prove the contrapositive form. Suppose M" is T For

a l l x e M , {x} i s closed so M \ {x} is open in the quotient

topology. Thus JI=, (M -» {x}) is open in L'M . Hence

VH N I t , (M \ {x}) = l £ ,U} is closed. •

Remark. I t i s known (of. 1.1.3) that if M is T in the semimetric~~~~ 1

p-topology then M i s T and p i s a metric.However, our previous

result does not preclude the existence of sets in M open in the

quotient topology hut not open in the p-topology.

It is vorth pointing out that from 2.6 we also obtain a homeomorph

of M if ve use the quotient by the holonomy group of the completed

holonomy bundle. Hence the current and subsequent results concerning

M have similar statements for that situation; see the example after

3,k belov.

M is E, •»=» graph G is closed. M is Kausdorff if and only if

graph G is closes in I/H .

Proof. (Cf. Kelley [16], p.98, Eells [10], p.6l.) The method is

interesting. Hote first that graph G is the equivalence relation

| u £ L'M•8

and that M = iTlJ/A . Also, the graph of L-

D = {(x,x) | K M } ,

i s an equivalence relation vith M/D = 55 and

{(u,R (u))•8

git*)

, n-L,(v)

i) Suppose M is Hausdorff. If (a,b) A then there exist dis-

joint neighbourhoods U of !!=_, (a) and V of H=-, (b) open in

M . (We use the quotient topology of course.) It,(

are therefore open in L'M . Moreover, no point of one lies on: a fibre which meets the other: they are not A-related. Hence

Jig, (U) * IU, (V) is an open neighbourhood of (a,b) , disjoint

from A , So the complement of A is open and A is closed.

ii) Suppose A is closed and 11 , (a) 4 IIj-,(b) & M . Then (a,b) \ A

But A is closed so there are open neighbourhoods U ,V of a,b

*• respectively vith no point of I) in a fibre meeting V : they

are not A-related. Hence I1=,(U) and Il^.tv) are disjoint

-1*7-

II. 3.1*

and open neighbourhoods of 11=, (a) and JI=, (b) because by 2.2

Us-, is an open map. •

Corollary. If M is Hausdorff then D is closed in M x M .

Conversely, since the identity relation gives rise to an open projec-

tion M •* M/D = M , if D is closed then M is Hausdorff. «

Sxample. The holonomy bundle through (xQ,b ) 6 L B with constant

connection X ¥ 0 has a completion that by 1,5.7 and II.2.6 is

uniformly equivalent to

{(XxQ + log bQ,v) | v > 1/|A|} .

The structure group * is trivial so we have M = (-°=,1/|A|] .

Furthermore, if A is closed then $(A) • A is closed so graph $

is closed; we expect M to be Hausdorff. Clearly this is the case

for (-»,1/|X|] .

B|y taking U x + log bQ)modX we find the completion of the holo-

nomy bundle through (xQ,b0) « L S vith constant connection X f 0 .

In this case, however, the holonomy group * Is the integers (cf.

1.5.7} and therefore non-trivial. The orbit of t is an exponentially

distributed copy of the Integers, which is not closed (see belov) and=1

so by 3.2 S is not expected to be T.

b-boundary point1 ' Consider the solitary

, described in 2.6, Let A be an open set in

the quotient topology for S , vith SI * A . We shall see that A

must contain all of S because the fibres in L 5 (x^b ) over A

must contain a family homeomorphic to

{[{(Xx0 + log b0)modX,v)] | u > O 1/[A|}

for some real a > l / | x | , But log(v | \ | - l ) •* -» ,so as ve

have observed before a l l fibres over S will be met by the inverse

image of A . {Cf. 1.1* and [7].) We have seen that each holonomy

bundle here appears as a vertical line v t E1/|A|,<=) on the

cylinder S * [l/|X|,<») in the standard metric. A typical closed

set in this line is therefore [ l / j i | , a ] for some a > l/\x\ . I t s

image by the integer group $ i s :

Ukt 2

[l/|X|,aeXk+

which is not closed.

I I . 3 . 5

3.5 Incomplete fitirea in the full metric. Suppose that (u ) is a Cauchy

sequence in L'M without H a l t there and such that II. ,(u ) is con-Li n

tained in a compact subset of M . Then we have:

i) Some xQ £ M : II ,(x.) is incomplete in the full metric;

ii) M is at most T Q .

Proof. (Schmidt [21] established this result; we amplify his proof.)

l) By compactness in M there exists an infinite subsequence (u')with H ,(u1) convergent to some x in M . Now if (u')

L n n

eventually lies in nLi(x) then by £.3 it has a limit there if

we measure distances along curves wholly in this fibre. Here,

however, we use the full metric. Suppose that H (x) is complete

in this metric- Then it is closed in L'M , and (u ) has a

limit in nL,(x) if it eventually lies in this fibre. By hypo-

thesis this is not so and hence we may suppose that for some S <• S

a(u',IL", Cx}) > 0 for n > K .XX 1>

We can find v t IL,(x) such that for n > N

as n •+ Therefore,x =*• d(u' ,v )n n

(v ) are In the same equivalence class of the Cauchy

Now, llm !L , (u ' )IJ n

) are

completion; (v } i s Cauchy because as n.k

If follows that

lim (v ) = lim (u1 ) t l i -• L'Mn n

so nT,(x) is not complete "because IT. , (limCu')) = x .

ii) From i ) , H , (x) contains a Cauchy sequence (v ) without limit

there. Suppose lim (v ) = v * L'M - L'M . Then for all

neighbourhoods U of v we have

unn*,(x} 4 t

because v l ies in the boundary of II , (x) . So if V is openin M with 11 , (v) £ V then also x e V so H is at most TQ .

Example. Consider L S with constant connection X ¥ 0 . Evidently

S is compact and contains the projection of the Cauchy sequence

-1*9-

II.3.T

(vn) : W •* L V : n * ( ( X Q _ n)modl,eXn) .

That this is Cauchy is clear because i t l ies on the finite (horizontal)

curve in case 2, Example 1.1, and so

d ( v V « I f e'xt at I = I e"Xk - e'Xni/ixi •n

We know that the essential part (cf. 1,3 and [T]) of L S is uniformly

equivalent to

W = {(u,v) | u e S1 , v > 1/|A|>

with the standard metric. There our Cauchy sequence appears as

which has limit

modX,(l

Ux modX,l/Ji| ) £ W - W .

Now as we see from its definition above {v ) does lie wholly in the

fibre H (x ) for any x * S , so every fibre of L S is in-

complete in the full metric. Schmidt [20] gave an example of in-

complete fibres arising from a two-dimensional Lorentz manifold.

3.6 M is geodesically complete if L'M ia complete

Proof. (Cf. Schmidt [21].)

Suppose that (L'M , < , > ) is a complete Eiemannian manifold. Then

every Cauohy sequence is convergent in L'M , in particular so are

those on horizontal curves (cf. 2.5). In consequence any horizontal

curve of finite length has endpoints in L'M ,

From 1.5.5 a curve in M is a geodesic if and only if It ia the

projection of an integral curve of one of the standard horizontal

vector fields. By hypothesis these are complete so geodesies in M

can be extended to infinite parameter values. •

Bemark• The converse ia false. Geroch [12] gave an example of a.

spacetime manifold that was geodesically complete hut which contained

an inextensible timelike curve of bounded acceleration.

3.7 If (M,g) Is a Eiemannian manifold then its Cauehy completion is

homeomorphic to its 'bundle-completion with the Levi-Civita connection.

Proof. (Cf. Schmidt [21].)

He work with the bundle 0 M of orthonormal frames and the distance

-50-

3.8

function d induced on it by inclusion in the Riemannian manifold

(L M, < ,> ) . We also have a distance function d_ on M by

1.2.It, It is sufficient to show that for all x,y € M

Suppose c Is a curve in M with horizontal lift c* to 0 M . Then

we have (ef. 1.2)

|c(t)|g

tecause © effectively expresses c with respect to an orthonormal

basis and in such component3 | |^ appears as the standard norm on

Thus c has the same length as c If c* i s a non-

horizontal curve in 0 M which also projects onto e then i t s lengthwill be greater, because the conneetion-form-term [w(c*{t))f Is notthen zero. The metrical equivalence of c and ef is independentof the choice of orthonormal frame through which the l i f t is made.The fibres are equidistant throughout their extent and the resultfollows. •

As pointed out by Schmidt,the coincidence of M with theCauchy completionwhen i t is available does encourage the view that i tis the natural choice for general manifolds with connection. Certainlythis seems to be true for pseudo-Riemannian cases for i t is a recentresult of Stredder (cf. [6]) that the metric tensor field determinesthe Levl-Civita connection merely by requiring that i t constitutesno additional structure, in the sense that i t is natural with respectto res t r ic t ions . For the present we use 3.7 In establishing thefollowing proposition and corollary,

M need pot be locally compact

Proof. (Schmidt [21].)

Let R have i t s standard metric structure and define the subset

A = {(x.sin 1/x) 0} U {(0,y) 1}

oThis A i s closed so M = R *- A i s a Eiemannian manifold; we denote

by M~ the connected component of (0 , -2) t M . Ely 3.7 we can

effec t the Cauchy completion and find the b-boundary

3M~ = { (x . s in 1/x) I x i i O J U {(0 , -1)} .

-51-

3.9

II.3.9

Therefore M = M U 3M is not locally compact because (0,-1} has

no compact neighbourhood. •

Corollary. (Hawking and Ellis [15], p.263.)

Whereas the origin (0,0) is in the topological boundary M~ of M~there is no curve In M" with endpoint there and so the origin isnot In the b-boundsry (cf. 1.1.12). •

Remark. Consider the Eiemannian submanifold

M = E2 s. «0 ,y ) | |y| < 1} .

For each (0,y) with 0 < y < 1 there are tvo points in the b-

boundary 3M . Hote that the manifold distance structure induced by2

the standard Eiemannian metric on E differs from the usual topologicalmetric, since i t involves an infimum over curves in M . Hence thedistance of (l /n,0) from (-l/n,0) tends to 2 , for the minimisinggeodesic must pass through (0,1) or (0,-1) . In contrast, theEuclidean distance between the above points is /2/n which tends tozero.

b-incompleteness of curves in M . A curve c in H is said tohave finite bundle-length if i t has a horizontal l i f t c of f initelength in L'M . This attribute is independent of the choice ofpoint in TT^Ce) through which the l i f t is effected. For we have theproperty 1.5.1 i i ) of connections which assures us that the action ofG maps horizontal curves among themselves and t ransi t ivi ty guaranteesthat the process is surjective; unifora continuity as displayed in 1.3preserves the finiteness. (Cf. [15], p.259.)

A curve c : [d,l) •+ M is called b-incomplete if i t has finitebundle-length and admits no continuous extension in M to domain[0,1] , Evidently, the definition extendB t r iv ia l ly to any (piece-wise-C ) reparametrisation of the curve. From 2.5 i t is clear thatthe b-boundary consists precisely of the endpoints in M of b-incomplete curves in M .

We have seen in 3.5 that a non-convergent Cauchy sequence in L'Mwhose projection is trapped In a compact set of M implies incompletefibres and consequential loss of separation fac i l i t ies in M . Herewe give a formulation in terms of b-incomplete curves, due to Hawkingand El l i s .

-52-

3.10

3.11

II.3.11

Imprisoned b-incoErpleteness. A point x e M Is not Hausdorffseparated in M from a point y <• 3M if there is a b-incompletecurve c in M which has x as a limit point and y as an endpoint

in M .

Proof. (Hawking and Ellis [15], p.289. )

For the given curve c there is a horizontal lift c with an end-

point b which by hypothesis is such that

h « lC,(y)c L'M N L'M .

Let V be an open set, containing y , in M . Then nfl{v) isLi

open in L'M ; since it contains b it also contains c (t) for all

t greater than some t . But then all points c(t) for t > tm

must lie in V . Hence

x is a limit point of o

meets every neighbourhood of xm

because

We see that this type of boundary point generated by im-

prisoned b-incompleteness is qualitatively different from the situationof supplying points that previously were omitted when M was part ofa larger (Hausdorff) manifold. Hawking and Ellis have pointed out

that in Taub-NUT spacetime (cf. [15], p.289, pp.lTO-8) there exist

b-ineomplete null geodesies total ly imprisoned in compact sets .

A b-boundary contains the topological boundary. Let U be an open

submanifold of M such that i t s closure Uc is compact in M . Then

U & 3U .

Proof. Since U is a submanifold, L'U, VU and 3U are welldefined. We may as well suppose U f jK . For a l l x i U » UC ••> int(U)we have x4U ^y the openness of U (cf. 1.1.12). For the samereason ve can find a curve c : [0 , l ) -*U with endpoint x .

Choose any b_ £ njT,(c(O)) and through it construct the unique

horizontal lift cf . The latter curve possesses an endpoint

b,s IT" U) & L'UL

by compaatnees. Hence for all large enough n i l , 0* eventually

lies in the open ball

I/U

-53-

II.3.12Therefore, we can connect b^ to same point; on c^ in L'U lay a

minimising geodesic of finite length. The projection of this

geodesic is a b-ineomplete curve in U and since i t s endpoint is x

we have x « 3U . Therefore U £ 3U , *

Remark• Schmidt [22] has shown that every point in a spacetime mani-

fold (cf. I l l below for definition) has an open neighbourhood U

such that U = 3U . This property-,, called local b-completeness. i s

a non-trivial consequence of the fact that every point in a spacetime

has a normal coordinate neighbourhood in which no geodesic is

imprisoned (cf. [18], p.

3.. 12 M is b-complete if L'M is complete.

Proof. Suppose that c : [0,1)-* H is a b—incomplete curve. Let

c* be i t s horizontal l i f t through same 0. * IiT^(c(0}). JTow c f

has f ini te length in L'M but by continuity of IL , i t haa no end-L

point there. Hence it contains a non-convergent Cauchy sequence.

Thus if M is not b-complete then L'M is not complete. •

Example. Hawking and Ellis C[15],p.278) proved that i f M 1B aspacetime then a converse is also true (cf. I I I . l . l ) : a spacetimeM is b-complete i f and only i f OM i s complete. This converseto 3.12 depends on the afore-mentioned local b-completeness that isenjoyed by spacetimes by virtue of their Lorentz metric. Schmidt[22] pointed out that on R there i s a connection with respect towhich no point is locally b-eomplete. Namely, that V which in the

"IP ?standard chart has components at (1 ,1 )( B given by

11 otherwise T. = 0 .

The required property was established by the following geometrical

argument, clearly of considerable power.

i) <R is simply connected so by I-5.T for all u t L B we have

3>(u) = $ (u) .•

ii) 7 is analytic on the frame bundle and thus, by Kobayashi and

Nomizu [18], p.153, * (u) is determined by the successive

covariant differentials of the curvature tensor {cf. 1.5.lit):

H, VR, V2H,..., at II ,(u) .

iii) At the origin in R we find VE

vanishing components of R are

-54 -

0 and there the only non-

lv)

Hence

given

the

by

action of

II.3.12

2

$(u) on

v •+• eav|a

1

tan II ,(u) i s

We can choose a closed curve through the origin on which the

parallel transport corresponds to H with e™ < 1. Then for

successive circuits of this curve the horizontal l if t has

monotonically decreasing tangent vector. Hence for infinitely

many circuits the bundle-length is finite and ve have a b-incomplete

curve. The process is applicable at all points. Moreover,

arbitrarily small neighbourhoods can contain b-incomplete curves,

implying contributions to their b-boundary that have no counter-

part in their topological boundary.

By way of illustration here, we shall calculate the holonomy group

at the origin and display the b-incompleteness. Consider the

following closed curve c , consisting of four parts ,

c. a l l with domain [0,e] for some i. > 0

c , c . c and

c : <

t ~*(t,0) c1(t) = (1,0)

=2(t) - (0,1)

c3(t) = (-1,0)

cu(t) = (0,-1)

U image of c i s a square of side f. , with f i r s t corner

c1(0) * (0,0) and lying in the upper i lght-hand quadrant of R .

Evidently i t i s closed and homotopic t o zero. Let (A3 + B3 ,2 •

C31 + D3 ) be a bas i s for T, ,R ; we find cT , the hor izontall i f t of e through t h i s point in L E

The required curve i s given by (cf.I.S-1*)

: t *

where the matrix of functions y^ : [0,t] •* B satisfies

- 5 5 -

II.3.12

A B-

C D

We deduce the following so lu t ions , for the four p a r t s ,

: tB

C D

c» : t * C2(t),

o* : t » C3(t),

A Be

t t

C2l

Hence the corresponding element of * = K is e , for each E > 0.'.

(The inverse element corresponds to a curve with the same image as c

but having a clockwise sense of direction round the square, instead

of anti-clockwise.)

The bundle-length of c is the length of c , which is the sum

of the lengths of the four parts:

1 . 0

f.1.2)

where (S^l = (Xjj)"1 • Plainly this length is finite; ve denote

it by LQ .

How consider the curve c,. which is the countably infinite

composition of c with itself, corresponding to an unending series

of circuits round the image of c . From above and the definition of

the bundle metric in 1.1 i t is easily seen that the bundle-lengths

of successive circuits decrease with constant factor e = r , say.

It follows that the bundle-length of c . is

L o r

k = 0

-56-

II.3.13

which exists and is finite for 0 < r < 1, by the property of2

geometric series. Thus c. is a t—incomplete curve,in Ft with thegiven connection. Finally for any neighbourhood of the origin in

•R we can so choose £ > 0 that c lies in this neighbourhood.2

Likewise for other points in R . We note that, rather like the

situation in Example 2, l.k, the horizontal l i f t of c is an

exponential spiral up the frame bundle, albeit on a square base here,

3.13 Projection of finite incomplete curves. The projection of a finite

incomplete curve in L'H is a curve of finite bundle-length.

Proof. We suppose that L'M is incomplete and that there exists some

(piecevise - C , as usual) curve

c» : [0,1)-+ I'M

which admits no continuous extension to domain [0,1] and which has

finite length. We shall take the horizontal projection of c* and

show that i t is the horizontal l i f t with finite length of the

projection

nL,»c* - c : [0,1) -*M .

The property 1.5.1.1} of connections gives us a smooth

decomposition of the tangent vector to the curve c*

Therefore we have a piecewlse-continuous horizontal vector field

c_ along e* . By the fundamental theorem for ordinary differential" .* 1

equations the field.*c

. 1c_ has a unique piecevise - C integral curve

with i = c " and cH(o) = c*(0). By our construction i t is

* :clear that c,, Has the same projection, c, as cil

IL,»c = IL ,«c* = c : [0,1) —r« .

Saw, every horizontal vector field on L'M is the horizontal lift

of same vector field on M (cf. 1.5.3-v)). Also we have from the

isomorphism property in 1.5.1

c(t)

-57-

II.3.13

which by I.5.1* ensures that c is precisely c the uniqueit

horizontal lift of e through c*(0), perhaps with some constant

sections included, where c* was vertical.

The bundle-length of c is (cf .1.1. , 1.2.1*)

|c\t)(dt at

which cannot exceed the given finiteness of

rl|ctt)|dt

because

|c*(t) | 2

Thus e, the projection of the finite incomplete curve c* , has

finite bundle-length. •

Bemark. To stre»gthen our result to a full converse of 3-12 we seem

to need more structure than a mere connection. For example, Hawking

and Ellis [15] have shown that in a spacetime a projected curve of

finite bundle-length has no endpoint and is therefore b-inccnplete.

Their proof of the corresponding result to 3.13 differs from ours

(cf.111.1.l). The sufficiency of a spacetime in providing the

extra structure required fcy Hawking and Ellia is implied by a lemma

of Schmidt [22]. We can see whaj. is needed by the following

argument.

We try to contradict the supposition that c , in the proof of

3.13, has an endpoint x in M . Thus there is a continuous

extension of c

c: [0,1]-+ M , with c(l) = i ,

such that for all neighbourhoods H of x there exists t t[0,l)

with

c(t) t H for t > t

-58-

II.3.13

We knov that II*",(H) i s open in L'M and that i t contains c*(t)

for t > t x . If they exist in L'M , the limits

lim cT(t) and Urn c»(t)t + 1 t + 1

must lie in H-^tx) vhich is contained in HJ*", CH) . The Lorentz

metric structure on a spacetime allovs us eventually to trap c*

and c* in a compact subset of II*,(H) and so guarantee the

existence of the limits to contradict the condition that c" is

without endpoint.

-59-