CONNECTIONS by ROBERT ALEXANDER NICOLSON B.Sc, University of B r i t i s h Columbia, 1 9 6 9 A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF SCIENCE i n the Department of MATHEMATICS We accept t h i s thesis as conforming to the required standard THE UNIVERSITY OF BRITISH COLUMBIA APRIL, 1 9 7 2 In presenting this thesis i n p a r t i a l fulfilment of the requirements for an advanced degree at the University of B r i t i s h Columbia, I agree that the Library shall make i t freely available for reference and study. I further agree that permission for extensive copying of this thesis for scholarly purposes may be granted by the Head of my Department or by his representatives. I t i s understood that copying or publication of this thesis for financial gain shall not be allowed without my written permission. Department of l7j^L^mo3xA/) / The University of B r i t i s h Columbia Vancouver 8, Canada Date ttfal 7 if 72 [ii] Supervisor; Dr„ J . Gamst ABSTRACT The main purpose of t h i s exposition i s to explore the r e l a t i o n s between the notions of covariant derivative,, connection, and spray. We begin by introducing the basic d e f i n i t i o n s and then use a method of Gromoll, Klingenberg and Meyer to show that covariant d e r i v a t i v e s and connections on vector bundles are i n a natural one-to-one correspondence. We conclude by showing that on the tangent bundle of a manifold, sprays and "symmetric" connections are i n a natural one-to-one correspondence. Although we use a d i f f e r e n t method, t h i s re-establishes a r e s u l t of Ambrose, P a l a i s , and Singer. TABLE OF CONTENTS 1 Introduction Chapter 1. 1.1 1.2 1.03 1.4 1.5 lo6 1.7 BUNDLE CONNECTIONS 3.2 3.3 21 Tangent Bundles of Smooth Groups 21 The Double Structure of TE 31 D e f i n i t i o n of TT 34 The Kernels of Tf' 35 Connections on Vector Bundles 39 The Difference F i e l d of Two Bundle Connections 44 Chapter 3* 3.1 6 Definition 6 Local Coordinates 7 P a r a l l e l Transport 11 The Difference Between Two Covariant Derivatives 13 Covariant Derivatives on Manifolds 14 Geodesies 16 Difference and Torsion 17 Chapter 2. 2.1 2.2 2.3 2.4 2.5 2.6 COVARIANT DERIVATIVES COVARIANT DERIVATIVES VERSUS BUNDLE CONNECTIONS From Connection Forms to Covariant Derivatives 4# From Covariant Derivatives to Connection Forms 52 Differences 54 4# Chapter 4« SPRAYS „ 4.1 Preliminaries 55 4.2 Bundle Connections on Manifolds 57 4.3 Second Order D i f f e r e n t i a l Equations and Sprays 60 4.4 The Connection Map of a Spray 63 55 Bibliography 65 Appendix 66 1 INTRODUCTION One of the main objects of interest to the d i f f e r e n t i a l geometer are the geodesies of a Riemannian manifold. Recall that, i n l o c a l coordinates, geodesies are solutions of a system of second order d i f f e r e n t i a l equations? oy(t) + r [ ^ ( < r ( t ) ) oj'(t) oj'(t) where the \^ . are the c l a s s i c a l C h r i s t o f f e l = o symbols. To simplify the notation, we introduce, f o r x i n some coordinate domain, the b i l i n e a r t given by I^( i» j) e e lR - n X TR >TR 2 ^(x)e^, where {e^\ n the standard basis of lR „ n denotes In these terms, the above n d i f f e r e n t i a l equation reads: <r"(t) + Q-( )( c r ' U ) , cr»(t)) t « 0. Of course, the V\ 's depend on the l o c a l coordinates one uses. The basic d i f f i c u l t y encountered by the founders of the theory i s that the Hi' form as the components of a tensor". 8 n o t trans- That i s , the Q do not define a b i l i n e a r map on tangent spaces. I t was L e v i - C i v i t a who saw that the b i l i n e a r maps •s 2 P' do have an i n t r i n s i c meanings they allow one to introduce absolute d i f f e r e n t i a t i o n port along curves. CT~ s • I and p a r a l l e l trans- We r e c a l l the r e s u l t : f o r curves -» M on a Riemannian manifold M one considers vector f i e l d s to along c r , that i s , " l i f t s " of cr to eu'rves CO : I > TM on the tangent bundle TM of M. associates with to Absolute differentiation a further vector f i e l d K^co along cr. In l o c a l coordinates: ; co t i >u> »(t) + C - ( ) ( CT'Ct), to ( t ) ) . t i s c a l l e d a p a r a l l e l family along CT" i f ^.ui = 0. So, i n l o c a l coordinates, p a r a l l e l f a m i l i e s are the solutions of a homogeneous l i n e a r d i f f e r e n t i a l equation. Therefore, f o r each curve cr on M, the p a r a l l e l f a m i l i e s along c r form a vector space P,,- , and evaluation at any t« i n the domain of cr gives a l i n e a r isomorphism of P<j- with the tangent space of M at c r ( t ) . 0 Put differently, p a r a l l e l f a m i l i e s give a s p e c i f i c way of propagating tangent vectors along curves. In terms of p a r a l l e l f a m i l i e s , t r i s a geodesic i f and only i f t r ' i s a p a r a l l e l family along 0 ~ . Moreover, one can use " p a r a l l e l transports" along i n t e g r a l curves of a vector f i e l d X to define a general covariant deriv- 3 ative of vector f i e l d s Y with respect to X. Considerable e f f o r t has been spent i n the l a s t fifty years to f i n d i n t r i n s i c formulations of the various aspects of the foregoing theory. The idea of absolute d i f f e r e n t i a t i o n leads to the notion of a covariant derivative on a vector bundle; that i s , an operator taking a vector f i e l d X on a manifold M and a section S of a vector bundle E over M to a further section V S of E. X In Chapter One we review the basic f a c t s about covariant d e r i v a t i v e s and show how one extracts the analogue of the C h r i s t o f f e l symbols from the formal d e f i n i t i o n . The problem of formulating the idea of p a r a l l e l transport along curves turns out to be more subtle. One might be tempted to specify, f o r each curve (T on the manifold, the vector space of a l l p a r a l l e l f a m i l i e s along o- . However, t h i s i s not how p a r a l l e l transport a r i s e s i n p r a c t i c e . Moreover, i t i s t e c h n i c a l l y d i f f i c u l t to formulate smoothness conditions i n such a context. The way out i s to consider the " i n f i n i t e s i m a l " aspect of the s i t u a t i o n . tangent vector In other words, one prescribes f o r each ^ at a point x on a manifold M, the vector space o f . a l l " i n i t i a l v e l o c i t y vectors" of p a r a l l e l f a m i l i e s along curves which pass through x with "velocity" ^ . Note that p a r a l l e l f a m i l i e s are curves on the tangent bundle, hence t h e i r " v e l o c i t y vectors" are i n the tangent bundle of the tangent bundle' So, one i s forced to consider the "double tangent bundle" of a manifold. Actually, one gains i n c l a r i t y by generalizing to the case of an a r b i t r a r y vector bundle. Accordingly, i n Chapter Two we start by analyzing the structure of the tangent bundle TE of a vector bundle E over a manifold. The d i f f i c u l t i e s with the f a i l u r e of the C r i s t o f f e l symbols to "transform l i k e a tensor" show up again: TE i s not a vector bundle over M. TE does, however, carry two d i s t i n c t vector bundle structures, one over E, and the other over the tangent bundle of M. We then give the formal d e f i n i t i o n of a connection on a vector bundle E over a manifold M: i t i s a map which assigns to each tangent vector ^ of M a subspace of the f i b r e of TE over ^ with respect to the vector bundle structure over TM. F i n a l l y , Ambrose, P a l a i s and Singer showed how one can deal d i r e c t l y with geodesies by introducing the notion of a spray on a manifold Ms i t i s a vector f i e l d on the tangent bundle bundle of M whose i n t e g r a l curves "look l i k e geodesies". In Chapter Four we r e c a l l the basic properties of sprays and show how one obtains the analogue of the Christoffel symbols. The main purpose of t h i s exposition i s to explore the r e l a t i o n s between the notions of covariant derivat- 5 ive, connection, and spray. In Chapter Three we use a method of Gromoll, Klingenberg, and Meyer to show that covariant derivat i v e s and connections on vector bundles are i n a natural one-to-one correspondence. In Chapter Four we show how, on the tangent bundle of a manifold, sprays and "symmetric" connections are i n a natural one-to-one correspondence. Thus, we r e - e s t a b l i s h the main r e s u l t of Ambrose, P a l a i s and Singer by a d i f f e r e n t method. 6 Chapter 1; Covariant Derivatives Section 1.1s Definition Let M be a smooth manifold of dimension n, and let p s E >M denote a smooth vector bundle over M with f i b r e K . For an open subset U of M, l e t k V(U) denote the vector space of smooth vector f i e l d s defined on U, and ^ ( E ) the vector space of smooth sections of E over U. DEFINITIONS V A covariant derivative on E i s an operator s V(M) X > T (E) > Vxs ^(E) M ( X ,S ) i having the f o l l o w i n g properties: Dl) V X Y =V S V S, S + X + Y D2) \7fXs D3) V (S+T) = V S + V T, and D4) V (fT) = (Xf)T + fV T, X x = fV S, x X X x ( where f i s any smooth real-valued function defined on M ). i 7 Section 1.2 s Local Coordinates In order to exhibit l o c a l coordinates f o r covariant d e r i v a t i v e s we f i r s t examine t h e i r r e s t r i c t i o n to open subsets of Mo I 1.2.1s R e s t r i c t i o n to Open Sets Let M be a smooth manifold of dimension n, and l e t »M denote a smooth vector bundle on M with p %E k _ fibre B . Let V be a covariant derivative on E, and l e t U be an open subset of M . LEMMA; (a) I f Y e V(M) vanishes on U, then VyS vanishes .. on U f o r each S e ^ ( E ) , and (b) i f T € fJj(E) vanishes on U, then \7 T vanishes X on U f o r each X e V ( M ) . PROOFt We w i l l prove only (b), f o r (a) i s even more straightforward. Fix some point y e U, and l e t f : M * IR be a smooth function, having support i n U, such that f i s i d e n t i c a l l y 1 on some neighbourhood of y. definition fT « 0 € t(E). Then by Thus 0 « V fT = x (Xf)T + f V T, x so, at 7, we have 0 = (Xf)(y)T(y) + f ( ) ( V T ) ( y ) 7 = x (V T)(y). x Since y € U was a r b i t r a r i l y chosen, the lemma i s proved, PROPOSITION: If V i s a covariant derivative on E, and U i s an open subset of M, then there e x i s t s a covariant d e r i v a t i v e ( X , S ) e V(M) X V on p~ (U) such that, f o r any rjj(E), ( V S)(y) = v V^S(y) f o r each y « U, where X € V(U) and S e the r e s t r i c t i o n s PROOF: f^CE) are of X and S r e s p e c t i v e l y . The problem i s to define V on vector fields and sections which may not be extendable to global vector f i e l d s and sections. So, l e t X be i n V(U) and S i n H^(E), and f i x some point y € U. as i n the proof of the lemma. Let f be' Then fX e V(M) f S € [^(E), so we may set (VjSKv) * ( V fS)(y). f x and 9 The lemma ensures that V i s well-defined, and by construction i t i s a covariant derivative on p""*( uns a t i s f y i n g the conclusion of the proposition. § E x p l i c i t Local Coordinates 1.2.2; Let M be a smooth manifold of dimension n, and let p ; E »M denote a smooth vector bundle over M with f i b r e Et . Let V be a covariant derivative on E, and l e t U be an open subset of M such that TM, the tangent bundle of M, and E are t r i v i a l over U. Note that V(U) may be i d e n t i f i e d with C^CU^R ), the 11 vector space of smooth ]R -valued functions defined on n U, and rj(E) with C*(U^!R ), by considering p r i n c i p a l k parts. The r e s t r i c t e d covariant derivative V thus induces an operator o U : C°°(U^R ) X n C~(U,IR ) k > C°°(U,JR ). k According to the d e f i n i t i o n i n [ 1 . 1 ] , t h i s operator SU i s l i n e a r over C^CUjJR) i n the f i r s t argument but not i n the second. I f , however, f o r F € C^Cu,©11) and G e C ^ C U ^ ) , we define T (F,G) U where = DG : U S (F,G) - DG(F) U > Lin(JR ,]R ) n k i s the d e r i v a t i v e of G, then one checks e a s i l y that T i s l i n e a r over C°°(U,E) i n both v a r i a b l e s . U 10 LEMMA: I f an operator T : C°°(U,IR ) X C°°(U ]R ) n > C~(U,JR ) k k 0 i s b i l i n e a r over C^CUpIR), then (a) i f F e G^U^IR ) vanishes at x « U, P(F,G) vanishes 11 at x f o r each G e C°°(U,lR ), and k (b) i f G e C°°(U^R ) vanishes at x € U, P(F,G) vanishes k at x f o r each F e G°°(U,JR ). n PROOF: We w i l l prove only the f i r s t assertion. For each i from 1 to n, l e t E^ € C°° (\J B ) 9 the constant function onto the i vector o f J R . n n canonical basis Then we may write F =s f.E, s i 1 1 where each f ^ e C^d^R) vanishes at x. any G e C denote Then f o r (U,JR ) we have k r(F,G)(x) = = = rCSfiEiyGXx) Sf,(x)r(E.,G)(x) 1 1 i 0. Returning to the e a r l i e r discussion, we can now see that f o r each x e U, P induces a b i l i n e a r morphism : JR X n JR k > JR . k 11 That is$ i f , f o r (dx,u) € IR X n IR , we choose F 6 k C°°(UjIR ) such that F(x) = dx, and G € C°°(U»IR ) such n k that G(x) = u, and set r^(dx,u) r (F,G)(x), = U then, by the lemma, P. i s well-defined; and, by construction, i t i s bilinear. Moreover, by d e f i n i t i o n the map p . TJ Bii(m x m , m ) n > Xl k k >rx i s smooth. So ? over U, V i s represented by the smooth family of b i l i n e a r P s m X TR > k n i n the sense that S (F,G)(x) U = DGl F(x) + T ( F ( x ) , G ( x ) ) . |x Consequently, x V-^S(x) i s already determined by the value of X at x and the values of S along any smooth curve f i t t i n g X(x). § 1.3s P a r a l l e l Transport Let M be a smooth manifold of dimension n, and let p s E >M denote a smooth vector bundle on M with f i b r e B^. and l e t cr ; I Let V be a covariant d e r i v a t i v e on E, »M be a smooth curve. By a s e c t i o n of E along o~ we mean a smooth 12 such that p « S s c r , section S of E along The covariant derivative of a c r i s the new section V S of E along c r —> E : I t defined as f o l l o w s : For f i x e d t e I, we note that the map 0 c* : t i »-(t + t) 6 c r i s a smooth curve representing Moreover, S i s defined along 9 '(t© ) = t| c ^ c r ( t so we may form V^S(t ), 0 which we define to be the value of V^S at t In l o c a l coordinates, i f V r "1 . • TO v^- n kl. . 0 i s given by 7D^ . UI. ir»^ M.L , and S by G one f i n d s that t , S : I »IR , k i s given by » G'(t) + r (cr (t), , < r ( t ) G(t) ). We say that a section of E along c r i s p a r a l l e l along c r i f V^S E 0, L o c a l l y , f i n d i n g sections p a r a l l e l along cr means solving G'(t) +£(t)(cr'(t),G(t) ) = 0 13 f o r G. Since t h i s i s a l i n e a r homogeneous d i f f e r e n t i a l equation, we see that p a r a l l e l sections along any given curve e x i s t , and are uniquely determined by any one of t h e i r values, § 1.4: The Difference Between Two Covariant Derivatives Let M be a smooth manifold of dimension n, and l e t p :E >M denote a smooth vector bundle over M with f i b r e E . Let V and V be covariant derivatives on E. Since I^(E) i s a r e a l vector space, the following i s well-definedo Let D(V,V) D(V,V) denote the map : V(M) X ^ ( E ) >^<E) ( x , s )i > V x s - V x s. C l e a r l y D(V,V) s a t i s f i e s conditions Dl) to D3) of [1.1] and condition D4) y i e l d s : D(V,V)(X,fS) = V (fS)-V (fS) « (xf)s + fVxs - (xf)s - fVxs • fV s x x x - f Vxs Thus the difference D(7,V) i s b i l i n e a r over C~(M R). f D(V,V) V and ^7 i s c a l l e d the difference tensor of . Suppose now that V i s any covariant d e r i v a t i v e defined on E, and that D J V(M) X T (E) M -> (E) r M i s b i l i n e a r (over C ^ M , © ) ) . Setting V i t i s t r i v i a l to check that V « V - D, i s then a covariant 14 derivative of E, and that D(V,V) = D. Locally, over a suitable coordinate domain U in M we may represent V 9 T u s C~(U,B ) X n by a b i l i n e a r map C°°(U,E ) > C°°(U,R ), k and V by a s i m i l a r map T . U i s then represented by T ' u k The difference D(V,vT which i s again b i l i n e a r over C°°(U,IR). Section l'»5i 1 1.5.1s Covariant Derivatives on Manifolds D e f i n i t i o n and Local Coordinates Let M be a smooth manifold of dimension n, and l e t TT s TM >M denote i t s tangent bundle. Recall that t h i s i s a smooth vector bundle over M with f i b r e IR . n A covariant derivative on M i s a covariant derivative on TM as defined i n [1.1], V s V(M) X V(M) ( X , T )» That i s , an operator > V(M) > \7 Y X s a t i s f y i n g conditions Dl) to D4). Let U be an open subset of M such that rT (U) 1 is trivial. Then, as i n [1.2.2], we may i d e n t i f y V(U) with C°°(UpIR ). n Let X and I be smooth vector f i e l d s on U with corresponding respectively,. maps F and G € C°°(tT,E ) n Then, i n the notation of [1.2.2]) we have, f o r x € U, g(F,G)(x) where the induced CI. DG(F)(x) + I^(F(x),G(x) ) map 3R n : = E X n HRn i s bilinear. Since V P derive an e x p l i c i t representation of *s, we may i s determined l o c a l l y by these i n terms of the coordinate system of U. § 1.5.2; Classical Notation We w i l l continue with the notation of the l a s t paragraph. Let E, denote the i vector of JR . n Then we may F where each = Then, since Q write . n,E. i i € C^CUjB), and G = canonical basis similarly g.E.. 2 3 3 3 i s b i l i n e a r , we have I^CFUKGCX)) = P( 2 f (x)E , 2 g (x)E ) i j x i 2 f (x)g.U) ± i tJ Moreover, each map rj\ : U i j j ^ (E ,E .). x i J >m n ^(E^E.) 16 must be smooth, so i t may be written where each € C°°(U,m). F i n a l l y we have r(F(x),G( )) = X x 2 f (x)g.(x)[ 2 1, j .k J so r , and hence V, n 3 t 1 J 1 rt(x)E.], K i s given l o c a l l y by s p e c i f y i n g smooth real-valued functions on U. Section 1.6: Geodesies Let M be a smooth manifold of dimension n, and l e t V be a covariant derivative on M (that i s , on the tangent bundle of M). Let cr : I a vector f i e l d along ^ M "be a smooth curve. cr we mean a section to : I By > TM of TM along c r . As i n [1.3] we say that oo i s p a r a l l e l along cr i f ^U> = 0. Note that the canonical l i f t of CT i s a p a r t i c u l a r vector f i e l d or ' : I * TM along c r . We say that CT" i s a geodesic with respect to V if G~ * i s p a r a l l e l along where V So, i n l o c a l coordinates, i s represented by T cr 0~~. x : E n X l n >B , n i s a geodesic i f and only i f 0 - " ( t ) +IJ (crHt),cr»(t)) (t) £ 0. In other words, to f i n d geodesies means to solve an 17 e x p l i c i t second order d i f f e r e n t i a l equation which i s quadratic i n 0*'. By the existence theory of ordinary d i f f e r e n t i a l equations, f o r each x e M, and each <f e T M.,„ there e x i s t s a unique geodesic cr • I > M such that cr (0) = x, ando-'(O) = cj . Section 1.7: Difference and Torsion < « Let M be a smooth manifold of dimension n, and l e t V and V be covariant d e r i v a t i v e s defined on M. As i n [1.3It t h e i r difference tensor D : V(M) X V(M) ( X , Y )i i s bilinear. > V(M) >V Y X ~ VY X In t h i s case, moreover, we may D into symmetric and a l t e r n a t i n g parts. decompose Thus we write D(X,Y) = S(X,Y) + A(X,Y) where S(X,Y) = (1/2)[D(X,Y) + D(Y,X) ], A(X,Y) = (1/2)[D(X,Y) - D(Y,X)]. and PROPOSITION: The following are equivalent: (1) v (2) VX (3) S, the symmetric part of D, 7 X and \7 have the same geodesies, a VX X f o r each X € y-(M), and vanishes. PROOF; C l e a r l y , (3) implies ( 2 ) . 0 = D(X,X) f o r a l l X e V(M), we 0 f o r a l l X,Y = S(X,X) obtain = e V(M), Conversely, i f S(X+Y,X+Y) = 2S(X,Y) since S i s symmetric. Thus, (2) and (3) are equivalent. To see that ( l ) and (2) are equivalent, we work i n l o c a l coordinates. desic f o r V By d e f i n i t i o n , a curve cr i s a geo- if +r ( )( cr"(t) c r t 0-'(t),cr'(t)) = 0, and a geodesic f o r V i f cr"(t) + f j r ( t ) ( c r U t ) , cr'.(t)) (where T and T represent x € M and ^ € T M, )• 0 Since, f o r each cr with tf(0) = y, see that (1) means: [^(uju) f o r a l l u €IR . V there i s a geodesic X and cr*(o) = <| we V and = = r^(u,u) Consequently, ( l ) i s equivalent to (2). n By the t o r s i o n tensor of a covariant d e r i v a t i v e V on M, we mean the T y : map V(M) X V(M) > V(M) defined by T 7 (X,Y) = V Y 2 - V X Y - [X,Y], 19 where [X,Y] i s the usual bracket of smooth vector fields. I t i s e a s i l y v e r i f i e d that T v i s bilinear and a l t e r n a t i n g over C (M,JR). 0 0 A covariant derivative V on M i s said to be torsion-free i f Ty s 0. Locally, i f X and Y i n V(M) are represented by F and G i n C°° (-U,-E) r e s p e c t i v e l y , then [X,Y] i s n represented by DG(F) - DF(G). by DG(F) Since V^Y i s represented +r (F,G), then we know T (X,Y) w i l l be given U v by the map X 1 U » JR >r (G(x),F(x)) - P (F(x),G(x)). x l Note that V n x x i s torsion-free i f and only i f each corres- ponding P. i s symmetrico By straightforward the computation, one may e s t a b l i s h following? LEMMA; Let V and V be covariant derivatives on M with t o r s i o n tensors and Ty respectively. Then i f A denotes the a l t e r n a t i n g part o f the difference tensor of V and V , we have T PROPOSITION: v - T^ = 2A. For any covariant derivative V on M, there e x i s t s a unique torsion-free covariant d e r i v a t i v e V on M having the same geodesies as V . PROOF; of V I f D = S + A denotes the difference and V tensor , then the conditions we want are T£ « 0 D = A. and Therefore, by the lemma, we must set V = V - (1/2)T . V One may show by computation that V derivative. i s the desired 21 Bundle Connections From the geometric viewpoint, i t i s desirable to characterize connections i n terms of morphisms of vector bundles. A d e t a i l e d digression on the vector bundles involved i s required i n order to accomplish t h i s . We begin by considering smooth groups and t h e i r associated tangent bundles. Section 2.1s Tangent Bundles of Smooth Groups A smooth group i s a smooth manifold having a compati b l e group structure. That i s , a structure under which m u l t i p l i c a t i o n and the taking of inverses are smooth Operations, we w i l l show that the tangent bundle of a smooth group i n h e r i t s a compatible group structure. The proof of t h i s i s greatly s i m p l i f i e d i f we express the d e f i n i t i o n of a smooth group i n the language of diagrams. § 2.1.1s D e f i n i t i o n of a Smooth Group DEFINITIONS A smooth group i s a smooth manifold G together with smooth maps in : GX i G * G G and : * G 22 such that? (l) the following diagram commutes ( a s s o c i a t i v i t y ) : G X GX id G X Q -» G X G m X idG G X (2) m m G -> G there e x i s t s a smooth map *• of the one-point manifold i n t o G such that the following diagram commutes (unit element): (id ,*0 Q -> G X id G (*,Id ) G G X (3) G G -> G , and the following diagram commutes ( i n v e r s e s ) : (id ,i) Q * GX G m G. Note that a l l of the maps appearing i n t h i s d e f i n i t i o n are smooth. We may therefor apply the tangent functor T throughout the d e f i n i t i o n and thus gain i n f o r - 23 matIon about the tangent bundle § 2,1,2; TG. The Tangent Group of a Smooth Group Let G be a smooth group and TG the associated tangent bundle. The following theorem shows that TG has a natural group structure compatible with i t s manifold structure, THEOREM: I f m u l t i p l i c a t i o n and inverses f o r G are given by maps m and- i respectively, then the tangent maps Tm and T i induce a compatible group structure on TG, Examining the diagrams of [2,1.1], we see that t h i s i s an easy consequence of the f a c t that the tangent functor T commutes with products. Thus we need only prove the following lemma. LEMMA: and TM X PROOF: I f M and N are smooth manifolds, then T(M X N) TN are naturally diffeomorphic. Let pr^ denote the canonical projection of M X onto M, and p r 2 the corresponding projection onto N. d e f i n i t i o n of M X The N ensures that these are smooth maps. Thus they w i l l have smooth tangents: Tpr N x s T(M X N) >TM Tpr~ : T(M X N) »TN. and 24 Together, these induce a smooth morphism of T(M X N) onto TM X TN Recall that a tangent v e c t o r ^ a t x € M may be represented by a smooth curve cr : I >M where I i s an open i n t e r v a l of IR containing 0, C"(0) = x, and cr-'CO) = f . Let o( represent a tangent vector at x € M, and p a tangent vector at y e N. M X Then, the d e f i n i t i o n of N ensures that there e x i s t s a unique curve f at (x,y) € M X pr^° ^ = pr °^ = N such that and 2 P This induces a morphism of TM X TN into T(M X which i s e a s i l y seen to be smooth. N) This new morphism i s inverse to the one introduced above, so the manifolds are indeed n a t u r a l l y diffeomorphic. We may now examine more c l o s e l y the structure of TG. Note that the map Tm ! TG X TG *TG i s given l o c a l l y by: Tm: ( (g,u), (h,v) )i If ©< represents u, and ^ >(m(g,h), T .m(u,v) ). represents v, then T •j m(u,v) 1 2 may be represented by $(t) = T 5 , where m( GL(t), £ (t) ). Thus on the tangent l e v e l , the m u l t i p l i c a t i o n comes from pointwi.se m u l t i p l i c a t i o n of curves. Consequently, the u n i t element, of TG w i l l be that vector i n the f i b r e of TG over the u n i t element of G which represents the curve at that point. § 2.1.3s constant That i s , the zero vector. Decomposition of TG Let G be a smooth group, and TG the tangent bundle. projection. Let p : TG >G associated denote the The group structure of TG may canonical be more ex- p l i c i t l y viewed under the decomposition to follow. Let m and i represent m u l t i p l i c a t i o n and inver- sion on G r e s p e c t i v e l y , and l e t Tm and Ti be t h e i r associated tangent maps. given i n [2.1.2] we p Thus p From the d e f i n i t i o n of Tm as derive: o Tm((g,u),(h,v)) = m(g,h) = m(p X p)((g,u),(h,v)). i s a smooth group homomorphism of TG onto G. Let OQ denote the canonical "zero-section" of That i s , 0 Q : G gi >TG » (g,0). TG. 26 Since 0 d T G i s represented by the constant curve at S g, and tangent m u l t i p l i c a t i o n i s e s s e n t i a l l y pointwise, 0^ i s also a smooth group homomorphism. p °0 G Moreover, i s the i d e n t i t y map on G. This s i t u a t i o n may be neatly described a l g e b r a i c a l l y i n terms of a semi-direct product. defined as follows. Let H and G be groups, and l e t p be a group homomorphism of H onto G. i Let K denote the kernel of p, and the i n c l u s i o n of K into H. Let s be a group homo- morphism of G into H such that p » s = i d ^ . The follow- i n g diagram describes the s i t u a t i o n : K- >H ^ G. ( P Note that f o r any h € plh-s.pdi" )) = p ( h ) p ( h ) ~ 1 1 = so h-s-pCh" ) = i ( k ) f o r some k € K. Thus h = i(k)s°p(h), 1 and as sets, H = i ( K ) X i s(G). I n terms of t h i s decomposi- t i o n , m u l t i p l i c a t i o n i s given by: - i ( k ) s C g ) . KkOsCg') - i(k)[s(g)i(k')s(g- )]s(g)s(g«) 1 K i s normal^ so, s(g)i(k *) s(g"*^") e i ( K ) , and so the product i s i n the desired form. I f we abuse notation, and l e t g e G denote the action g s i(k)i > s(g)i(k)s(g~ ), 1 27 then our formula becomes [i(k)s(g)]-[i(k')s(g')3 - [i(k)gi(k')]-[s(g)s(g')] A l g e b r a i c a l l y , then, H i s said to be the semi-direct product of G and K r e l a t i v e to the action of G on i(K) defined above. Returning to the e a r l i e r discussion and notation, we have that TG i s the semi-direct product of G with ker(p) r e l a t i v e to the action on ker(p) given by conju- gation with elements of OQ. 1 2.l.lj.: Reinterpretation of TG Let G be a smooth group with m u l t i p l i c a t i o n given by a smooth map m. Let e denote the u n i t element of G. Let TG be the tangent bundle of G, and l e t be as defined i n [2.1.3], We now p and OQ i d e n t i f y ker(p) and the action of OQ mentioned i n the l a s t s e c t i o n . C l e a r l y , as a manifold, ker(p) i s just T (G). Note that, as a vector space, T G has the structure of an additive group, and that there i s a canonical a c t i o n of G on T G e given as follows: For g 6 G, l e t int(g) : G •G be the inner automorphism h\ given by g. map at e, Since int(g) leaves >ghg~ e f i x e d , i t s tangent 28 ad(g) = T ( i n t ( g ) ) : e must be l i n e a r . >T G T G Q @ The r e s u l t i n g homomorphism ad s G >Lin(T G,T G) e e i s knovm as the ad .joint representation. THEOREM; I n the above notation; (a) the group structure on ker(p) induced by Tm i s vector space addition i n T G, and g (b) the a c t i o n of G on ker(p) induced by the semid i r e c t decomposition of TG i s the adjoint representation. PROOF: To see that (a) holds, we r e c a l l that the u n i t element of TG i s the zero-vector i n T G. Therefore, e since T m :T GX e e T G e »T G e i s linear, T m(^ , rr\ ) = T m( ^ , 0) + T m( 0, ^ ) e Q e (b) follows from the f a c t that i f ^ = c* +^ . € T G i s represented Q by a curve cr then ad(g)(^j) i s represented by the curve f ti >gcr(t)g"" . 1 Thus, i f o<v i s the constant curve with image g, we may rewrite the above map as t\— >^(t)cr(t)[oA(t)r 1 But, by d e f i n i t i o n , t h i s i s the curve representing - ^(g^OoU" ), 1 so we are done. 29 Note that ker(p), with t h i s structure, i s the additive group of the Lie-Algebra of G, denoted L(G). To sum up: semi-direct i f G i s a smooth group, then TG i s the product of G with the additive group of r e l a t i v e to the ad.joint § 2.1.5 s L(G) representation. Actions of Smooth Groups on Smooth Manifolds We define here what i s meant by the action of a smooth group on a smooth manifold. Let G be a smooth group with m u l t i p l i c a t i o n given m. Let M be a smooth manifold. We say that G acts smoothly on M i f there e x i s t s a smooth map *t ..: . G X M ->M such that: ( l ) the following diagram commutes: m X i dM G X G X M id G X G X M X Q M -» M, and (2) the following diagram also commutes: (*,id ) » G X M M C l e a r l y , i f G acts smoothly on M, then TG w i l l act by 30 smoothly on TM with the tangent action. 2.1.6: Special Case; GL^ The smooth group that we w i l l be interested i n i s GI» , the group of a l l l i n e a r automorphisms of TR . k k 2 As an open subset of IR , GL^ has a smooth structure which c l e a r l y i s compatible with the group structure. By d e f i n i t i o n , GL^ acts l i n e a r l y (and hence smoothly) on lR by: k GL X k TR > lR k (A , u) i k *Au. As a manifold, the tangent bundle TGL GL^ X simply i s V M , where M^ i s the vector space of a l l ( k x k ) k matrices. To i d e n t i f y the group structure of TGL^, we look at the "tangent action" TGL X k IR X k where we have i d e n t i f i e d TR — k >TR k T E with 1R X k k X TR , k E . k C l e a r l y the a c t i o n i s given by: (A,M,u,v)l Therefore, TGL ^2k c o n s fc >(Au,Mu + Av). can be i d e n t i f i e d with the subgroup of i t i n g of a l l matrices of the form s fA 0" M A A e GI^jM € M k 31 Note then that and OQ p A 0* M A i s given by -» A, by A A 0 0 A so the semi-direct decomposition i s K A o" "I A_ Section 2.2: "A o' _MA~ 1 1 0' • .0 (MA" ^). 1 A. The Double Structure of TE The tangent bundle of a smooth vector bundle i n h e r i t s two smooth vector bundle structures. 1 2.2.1: The Standard Structure Let M be a smooth manifold of dimension n, and let p : E > M denote a smooth vector bundle over M k with f i b r e TR • Since E i s a smooth manifold, i t has a tangent bundle which we denote n E : TE- »E. We c a l l t h i s the standard vector bundle structure of TE over E, or simply the E-structure. I f the t r a n s i t i o n function between two i n t e r s e c t i n g coordinate domains of M i s given by xi >-h(x) where h i s a diffeomorphism between open subsets of 32 TR , n then the corresponding t r a n s i t i o n f o r the tangent bundle TM i s given by: > (h(x) ,h (x)dx). (x,dx)i 9 Then since l o c a l t r i v i a l i z a t i o n s of E have t r a n s i t i o n functions of the form (x,u)i » (h(x),t(x)u), where t i s a smooth mapping of an open subset of TR n into GL^, the corresponding t r a n s i t i o n s with respect to the E-structure of TE w i l l be given by (x,u,dx,du) H > (h(x),t(x)u,h'(x)dx,t(x)du+t Note that "fibrewise", t h i s i s l i n e a r § 2.2.2: '(x)(dx)u). in(dx,du). The Tangent Structure Again l e t M be a smooth, manifold of dimension n, and l e t p : E with f i b r e TR . >M denote a smooth vector bundle on M We w i l l show that the tangent Tp : TE map > TM gives TE a smooth vector bundle structure over TM. I t s u f f i c e s to exhibit a system of l o c a l t r i v i a l i z a t i o n s of TE over TM i n such a way that the t r a n s i t i o n functions act l i n e a r l y on the f i b r e s of Tp. To do t h i s , f i x some point i n TE, and suppose that (U, <^ , p ) i s a vector bundle chart at i t s . image under n-g : T E — > E . 33 Then since p i p°* (U)^ 1 -» cp(U) X TR k i s a diffeomorphism, Tp s T(p~ (U)) >T(c£(U) X IR ) 1 i s also. k But we know that the tangent functor T commutes with products, and from the d e f i n i t i o n of Tp we get that TCp"" ^) 1 ) = Tp" (TU) 1 ? so (TU,T<j> ,T0) gives a l o c a l t r i v i a l i z a t i o n of TE. In [2„2.1] we saw that the t r a n s i t i o n f u n c t i o n s f o r t h i s system of l o c a l coordinates have the form (x,u,dx,du) i > (h(x) ,t(x)u,h '(x)dx,t(x)du+t '(x)(dx)u), where h i s a diffeomorphism of open subsets of TR , and n t i s a smooth map from an open subset of TR i n t o GL^. n I f we r e s t r i c t our attention to f i b r e of Tp over (x,dx), then we induce a morphism of the form (u,du) i * (t(x)u,t(x)du + t'(x)dxu). Note that the matrix representation t(x) 0 t'(x)dx t ( x ) . of t h i s morphism i s i d e n t i c a l to the image of the tangent map of t at (x,dx). This l i e s i n TGL^ which was estab- l i s h e d i n [2.1.6] to be a subgroup of G L . 2k So, the t r a n s i t i o n f u n c t i o n s do act l i n e a r l y on the 34 f i b r e s of Tp, and we hare a vector bundle structure on TE. We c a l l t h i s the tangent structure of TE over TM, or simply the TM-structure. Note that, although the t r a n s i t i o n functions are l i n e a r i n (u,du) and i n (dx,du) separately, they are not l i n e a r i n (u,.dx,du), and so we do not get a vector bundle structure f o r TE over M. Section 2.3? D e f i n i t i o n of Tf Let M be a smooth manifold of dimension n, and l e t p s E •—> M denote a smooth vector bundle over M with fibre B . Let : TM »M denote the tangent bundle of M. We denote by E X TM the submanifold of E X M s i s t i n g of a l l points (^,^) such that P(?) = TM con- ifc(j). Note that i f p*" denotes the projection of E X TM onto E, M then we have a smooth vector bundle over E with f i b r e JR , n and i f denotes the projection of E X TM onto TM, we have a smooth vector bundle over TM with f i b r e IR • Local coordinates f o r E X M structure. TM come from the product I n p a r t i c u l a r , points are denoted l o c a l l y by t r i p l e s (x,u,dx) and the projections by p* : (x,u,dx)i > (x,u) and TTJ^ s (x,u,dx) i > (x,dx). In [ 2 , 2 ] we saw that Tp s TE < »TM and TT e ; TE >E each e s t a b l i s h a smooth vector bundle structure on TE. Since the morphisms are smooth, we may combine them to obtain a smooth map (ng,T ) : TE >E X p TM Moreover, by d e f i n i t i o n , these morphisms agree i n the f i r s t coordinate, so the image of (TT^T ) w i l l be E X E p Denote t h i s new morphism TM. M T( : TE >E X TM M As an immediate consequence of the d e f i n i t i o n s , we see that l o c a l l y T l s (x,u,dx,du) » :—>(x,u,dx). Tf has two i n t e r p r e t a t i o n s . Considering TE and E X TM as vector bundles over E, TT i s f i b r e - p r e s e r v i n g M and acts l i n e a r l y fibrewise. Such maps are c a l l e d E-morphisms. S i m i l a r l y , considering TE and E X M as vector bundles over TM, Tt i s a TM-morphism. Section 2,ki TM The Kernels of Let M be a smooth manifold of dimension n, and l e t p s E > M denote a smooth vector bundle on M with 36 Let Tl : TE ~^E X M morphism" introduced i n [2.3]. f i b r e IR. TM be the "doubleThe following commutative diagram summarizes the r e l a t i o n s h i p s of the preceding section. E X TE TM <r M Since Tt i s both an E-morphism and a TM-morphism, i t has two d i s t i n c t kernels. 1 2.4.1: The E-Kernel of Here we consider TT E. TT. as a vector bundle morphism over That i s , we concentrate on the following part of the diagram: TT TE > E X M TM E PROPOSITION: morphism such that There e x i s t s a smooth vector bundle i : E X M E X E M E i TE -» TE E X M TM -> 0 i s a short exact sequence of vector bundles over E. Before g i v i n g a formal proof of t h i s proposition, 37 we analyze the s i t u a t i o n "geometrically". Let V e € E. denote the kernel of TT over E, and f i x some I f p(e) = x e M, the f i b r e s of TE and E X M over e are T E and T ^ r e s p e c t i v e l y . So, by d e f i n i t i o n Q of Tf , the f i b r e of ? E TM over e w i l l be the kernel of the tangent map of p at e. That i s , «= kernel of T p : T E Q Q -* T^M In other words, V". i s the tangent space at e to p™ (x) e —1/ E But p (x) = E i s a vector space, so Y~* may be i d e n t i - f i e d with E . We now exhibit the formal proof of the proposition. PROOF: Define i :EX E > TE M l o c a l l y by i s (x,u,v)i »(x,u,0,v). Examining the form of the t r a n s i t i o n maps of the E-structure of TE, we conclude that i i s a vector bundle morphism over E. Moreover, the l o c a l description of Tt ensures that i s a t i s f i e s the conclusion of the proposition. § 2.4.2: The TM-Kernel of Tt . Here we consider Tt as a vector bundle morphism over TM. That i s , we concentrate on the diagram 3* » E X : TM TM M TE TM PROPOSITIONf There e x i s t s a vector bundle morphism j i TM X M E TE such that 0 * TM X M E —^TE— *E X M >0 TM — i s an exact sequence of vector bundles over TM. Again, before giving the formal proof, we look at the s i t u a t i o n fibrewise. F i x l € TM, and suppose determine the f i b r e (TE)^ (TE)^ = n^C^) = x € M. of TE over <| . We first By d e f i n i t i o n Tp" ^). 1 Thinking of TE as the d i s j o i n t union of i t s f i b r e s over E we haves (TEL o 5 U (TE)« PIT E = U ' T p ( 6 )'fl e«E eeE -1 5 e 5 e TE -But Tp"* ^) n T E i s the inverse image of ^ under 1 g T p. g In particular, Tp^Cj) n T E = e p unless e i s over xs that i s , unless e i s i n the f i b r e E x of E over x. Suppose e i s i n E . x inverse image of Then Tp*" (t|) n T E i s the 1 g under T p Q and hence a t r a n s l a t e of 39 the kernel of T o , l a t t e r with E . (TEL In [2.4.1] we i d e n t i f i e d the Thus, as a set, U (TE) n T E = U E v = E X eeE s e cB = e . J e E . x Note that, a f t e r t h i s i d e n t i f i c a t i o n , 1f : TE acts on (TE)^ = *E X M E X x second coordinate. E TM by projection onto the x In other words, the f i b r e of the kernel of Tt over ^ may be i d e n t i f i e d with E . x Now, the formal proof of the propositions PROOF % Define j s TM X M E »TE l o c a l l y by j : (x,dx,u) \ •(x,0,dx,u), and check as before that j has the desired p r o p e r t i e s . Section 2.5s I 2.5.1s Connections on Vector Bundles Connection Maps Let M be a smooth manifold of dimension n, and l e t p iE >M denote a smooth vector bundle over M with f i b r e IR . k Let T f : TE >E X M TM be the "double mor- phism" of [2.3]. DEFINITIONS A connection map on E i s a smooth section C sE X M TM >TE 40 of TT which i s both an E-morphism and a TM-morphism. In terms of l o c a l coordinates, a connection map C must be of the form C : (x,u,dx)I— >(x,u,dx,f(x,u,dx)) where, since C i s an E^raorphism, f acts l i n e a r l y i n dx, and since C i s a TM-morphism, f acts l i n e a r l y i n u, when the corresponding f i b r e s are f i x e d . For each x e M then, C induces a b i l i n e a r map which we denote -T • IR XTR -P s (dx,u) « n k f c ——>TR k by >f(x,u,dx). The negative sign appearing i n the notation i s i n t r o duced so as to ensure l a t e r agreement with c l a s s i c a l notation. L o c a l l y then, a connection map i s given by C : (x,u,dx)i where > (x,u,dx, -UJ(u,dx)) > - f Z i s a smooth mapping of an open xi subset of TR into the b i l i n e a r maps from T R x m n I 2.5.2; n k to TR . Connection Forms There i s an equivalent formulation of [2.5.1] i n terms of connection forms. Let M be a smooth manifold of dimension n, and k 41 let p s E »M denote a smooth vector bundle on M with f i b r e IR . Let k C : E X TM , > TE M be a connection map as defined i n [2.5.1], two short exact sequences of [ 2 . 4 ] , That i s i It E X E »TE— =>E X TM M M over E, and 9 , TM X M over TM, R e c a l l the E — >TE — >E X M TM We may associate with C the two r e t r a c t i o n s ? K« : TEi > E X E M B v i a i„ and K mf : TE »TM X E M v i a j . These are given l o c a l l y by Kg s (x,u,dx,du)> >(x,u,du +[^(u,dx) ) and Krpjyj : (XpU,dx du)I y These agree i n the t h i r d coordinate. »(x dx du +f (u,dx) ) ? ? x Thus Kg and induce the same map K s TE- »E given l o c a l l y by K ;': (x;UpdX;du)i » (x,du + P ( u d x ) ). Note that K has the following properties: p K^ 42 CF 1 s K i s a vector bundle morphism over p : E — » M whose r e s t r i c t i o n to V i s projection onto the second coordinate, and CF 2 : K i s a vector bundle morphism over TT^ ? TM—>M TM whose r e s t r i c t i o n to V i s projection onto the second coordinate, where and V ™ denote the "E-kernel" and "TM-kernel" of TT r e s p e c t i v e l y . DEFINITION; A connection form on E i s a smooth map K s TE >E s a t i s f y i n g conditions CF 1 and CF 2 above. The one-to-one correspondence between connection maps and connection forms i s clear from the construction, so we are done except f o r some remarks on notation. We denote the connection form corresponding to a connection map C by K ; and, a l t e r n a t i v e l y , the connecC t i o n map corresponding to a connection form K by Cg.' L o c a l l y then, i f C i s given by C : (x,u,dx)i »(x,u,dx, - r ( u , d x ) ), then KQ i s given by K and i f K C : (x,u,dx,du)» K x i s given by K : (x,u,dx,du)* then C » (x,du + T ( u , d x ) ), >(x,du +P(u,dx) ), i s given by C R : (x,u,dx)» »(x,u,dx, ~ (x,dx) ). x 43 § 2„5.3: A l t e r n a t i v e D e f i n i t i o n of Connection Forms In Chapter Four we w i l l be t a l k i n g about geodesic sprays, and there a s l i g h t l y d i f f e r e n t d e f i n i t i o n of connection forms w i l l be used. Let M be a smooth manifold of dimension n. Let p : E — > M denote a smooth vector bundle on M with k f i b r e TR . map For any s e TR, l e t s also denote the smooth s :E *E given l o c a l l y by: s : (x,u)t Let »(x,su) s^. denote the corresponding tangent map . s TE • (x,u,dx,du)i *-TE Kx,su,dx,sdu). Then: PROPOSITION: A smooth map K : TE >E i s a connection form on E i f and only i f CF 1 i s s a t i s f i e d , and: CF 3 : K o* s f o r each s € R. «s s oK (Refer to [2.5.2] f o r the d e f i n i t i o n of connection forms on E, and CF 1.) PROOF: Suppose CF 1 and CF 3. K : TE s>E i s a smooth map s a t i s f y i n g Then, as we have seen, CF 1 implies that l o c a l l y K has the form K : (x,u,dx,du)l where »(x,du + [^(ujdx) ) P ^ U j d x ) i s l i n e a r i n dx. By CF 3> P^sujdx) = s l ^ U j d x ) , so, P„(u,dx) i s also l i n e a r i n u ( c f Appendix). Thus CF 2 of [ 2 o 5 . 2 j i s s a t i s f i e d , so K i s a connection form on E. The converse i s also e a s i l y established, so the proposition stands. Section 2 . 6 ; The Difference F i e l d of Two Bundle Connections The two characterizations of bundle connections i n [ 2 . 5 ] y i e l d two corresponding interpretations f o r the difference between two connections. § 2.6.1; The Difference F i e l d of Two Connection Maps Let M be a smooth manifold of dimension n, and l e t p :E »M denote a smooth vector bundle over M with k rv f i b r e IR . Let C and C be two connection maps on E as defined i n [ 2 . 5 . 1 ] and suppose they are given l o c a l l y by C : (x,u,dx)i >(x,u,dx,-P(u,dx) ) C % (x,u,dx)i ^(x,u,dx,-P.(u,dx) ). and Considering C and C as E-morphisms, t h e i r difference ^(CjSf) ; E X ^ TM »TE M - i s given l o c a l l y by Dg(C,C) ; (x,u,dx) r > (x VL O -V {\i dx.) 9 f t x f + I" (u,dx) ). x 45 The image of D (C,8') thus l i e s i n V , the E-kernel of E E ff , so under the i d e n t i f i c a t i o n of [2,2.4], we a r r i v e at an induced morphism given l o c a l l y by ( x u , d x ) l — ^ U p U y - P ^ U y d x ) + f^Cujdx) ) € E X s Similarly, - E. considering C and C as TM-morphisms, A/. Drj^CCjC) induces a map given l o c a l l y by (x,u,dx)i >(x,-P(u,dx) + r(u,dx),dx) «E X TM, These maps are both E-morphisms, and agree on the corresponding projections, so they both induce the same map D(C,C) EX TM *E M (x,u,dx) i »(x,-C(u,dx) + n,(u,dx) ). This new morphism we c a l l the difference f i e l d of C and cf. Note that the l i n e a r i t y requirements on ' and V ensure D(C,C) acts l i n e a r l y f i b r e w i s e . Suppose now that C i s any connection map on E, and that D : E X TM >E M i s a morphism as above. Then i f C i s given l o c a l l y by C s (x,u,dx) i D : (x,u,dx)i -> (x,u,dx,-r_(u,dx) ), and D by > (x,f (u,dx) ) , x 46 we can define a new C s (x,u,dx)i connection map C on E by —> (x,u,dx,-T (u,dx) + f (u,dx) ). /V I t i s e a s i l y v e r i f i e d that C i s well-defined, and i s the unique connection map s a t i s f y i n g : D(C,C) = D. I 2.6.2: The Difference F i e l d of Two Connection Forms Let M be a smooth manifold of dimension n, and l e t p s E — > M denote a smooth vector bundle on M with k . • **» f i b r e IR . Let K and K be two connection forms on E as defined In [2.5,2], and suppose they are given l o c a l l y by K : (x,u,dx,du) i ^ (x,du +P(u,dx) ) K : (x,u,dx,du) \ > (x,du * t^Cuydx) ). and There are two possible i n t e r p r e t a t i o n s f o r the difference between K and K. Considering them as morphisms over p : E >M, their difference D (K,K) E : TE > E (x,u,dx,du)t—>> (x, r.(u,dx)-r.(u,dx) ) A X E i s a vector bundle morphism which annihilates V , since f^(u,0) = u = T (u,0) x by d e f i n i t i o n . 47 Passing to the quotient space under the i d e n t i f i c a t i o n s of [2.5.2] we have an induced morphism given l o c a l l y by ± (x,r (u,dx)-r.(u,dx)) (x,u,dx)t v where ( x u d x ) e E X M 0 Similarly, TT^ t TM TM. 9 seen as vector bundle morphisms over B *M the difference m ( K $ ) annihilates TM V . This induces a morphism given l o c a l l y by »(x, r ( u , d x ) - r ( u , d x ) ) . (x,u,dx) \ c ? That i s , under our i d e n t i f i c a t i o n s , Drj^jCKjK*) induce the same morphism. Dg(K,ft;) and We c a l l t h i s the difference f i e l d of K and f , and denote i t simply D(K %). 9 F i n a l l y , note that i f C and (if are connection maps, and Kg and K~ are the corresponding connection forms, then D(K ,K{y) c = - 0(0,8:) where the l a t t e r i s as defined i n [2 6.1]. e The minus sign appearing i n the formula i s the r e s u l t of the e a r l i e r d e c i s i o n to define the P. 's to agree with more classical results. 4* Chapter 38 Covariant Derivatives Versus Bundle Section 3°1* § 3»l»ls Connections From Connection Forms to Covariant Derivatives Tangent Maps of Smooth Sections Let M be a smooth manifold of dimension n, and l e t k p 8E >M be a smooth vector bundle on M with f i b r e H . For a smooth s e c t i o n S 8 M >E of p, we denote by S # : TM ) TE the section TS of Tp. In l o c a l coordinates, S i s of the form S : xi where u E . >(x,u(x) ) i s a smooth mapping of an open subset of B I t follows that S^ i n the corresponding n into coordinate domains w i l l be given by s (x,dx)\ > (x,u(x) ,dx,u *(x)dx). The follovring property of such morphisms i s t r i v i a l PI s (S + T)^ = S^ + T^ There i s , however, another c h a r a c t e r i s t i c property. f be any smooth real-valued f u n c t i o n defined on M. that f o r a smooth section S of p, fS i s another section. L o c a l l y , i f S i s given by S 8 xi then f S i s given by fS s x i > (x,u(x) ) > (x,f(x)u(x) ). Let Note smooth 49 The tangent map, (fS)* ( f S ) ^ , must then be of the form : (x,dx)v—Mx,f(x)u(x),dx,f(x)u*(x)+f (x) (dx)u(x)). r This represents a section of TE as a vector bundle over TM, and so the right-hand side may be decomposed to give: (x,f(x)u(x),dx,f(x)u'(x)) + (x,0,dx,f '(x) (dx)u(x)). C l e a r l y , the f i r s t term of t h i s decomposition comes from applying f to S.^ . Examine the second term. Recall from [2.4.2] that the kernel of : TE » E X TM, M considered as a TM-vector bundle morphism i s i : TM X E M . > TE (x,dx,u)» >(x,0,dx,u). The second term of the decomposition l i e s i n the image of t h i s map. This means there e x i s t s a global section of pr-^ : TM X E » TM, which we denote dfS, such that l o c a l l y , f o r a suitable choice of coordinates, i o dfS : (x,dx) \ > (x,0,dx,f (x)dx,u(x)). v I f one abuses notation to i d e n t i f y (x,dx) i > (x,dx,u(x)) of pr^, (df)S simply i s the product of df with S. : TM S > IR with the section 50 This establishes the following property? P2 : (fS)* = fS n -8- i ( d f S ) i n the foregoing notation. 1 3olo2: The Action of S» on Vector F i e l d s Let M be a smooth manifold of dimension n, and l e t p s E f i b r e TR . >M denote a smooth vector bundle on M with Let S and T be smooth sections of p, and denote t h e i r tangent maps as i n [3»l.l]. Then i f X and Y are smooth vector f i e l d s on M, and f i s a smooth r e a l valued function defined on M, the following hold: S* -(X + Y) i) S* ii) o(fx) (S + T)* o iii) (fS)^o X iv) where i -• S ^ . X = x = + S*.Y, fS*«>X, ° X + T^° X. and «' f S * * X 4- i(XfS) i s the i n j e c t i o n defined i n [2.4.2]. The proof of the properties i s based on the f a c t that the f o l l o w i n g two diagrams commute: TE 51 where TT^ and p are the vector bundle s t r u c t u r a l maps E of TE over TM and E r e s p e c t i v e l y , and df i s the second —TJR component of Tf : TM 1,3"lo3 ' The ^ffiXE. Construction Let M be a smooth manifold of dimension n, and l e t p s E——^M k fibre E . denote a smooth vector bundle over M with Let K : TE >E be a smooth connection form on E as defined i n [2 5«2]„ a For a smooth vector f i e l d X on M, and a smooth section S of p, define VS X where S# = TS« K ° S* • X « That i s , the following diagram commutes: M By conditions CF 1 and CF 2 of [2.5v2]. and i ) to i v j of f [3»1»2], V i s a well-defined covariant d e r i v a t i v e on E, Note that over a f i x e d coordinate domain i n M, suit- able choices f o r charts give the following formulas: so a n d K s (x,u,dx,du)i »(x,du + HL(u,dx) ), X : > (x, | (x)), and S : xi S^»X VS X : xi : x i >(x u(x)): f xi > (x,u(x), t| (x) ,u'(x)|(x)), » (x,u •(*) <§ (x) + i^(u(x), <J ( )) x 52 Section In From C o v a r i a n t 3»2; Derivatives Forms we e s t a b l i s h e d [3«1] covariant derivatives. that ~ to Connection connection forms We now show t h a t induce one may go t h e o t h e r way. I 3«-2.1; Uniqueness L e t M be a smooth m a n i f o l d o f d i m e n s i o n n , a n d l e t p : E »M d e n o t e f i b r e IR . be e E , and s u p p o s e bundle over M w i t h p(e) = x a smooth s e c t i o n of p s u c h t h a t the we Fix e a smooth v e c t o r [3»1]» notation of e M. S(x) = e. f o r a n y smooth v e c t o r Let S Then, i n field X on M have (S*°X)(x) Let UV d e n o t e e That € T E. e the set o f elements o f T^E o f t h i s e form, is, u ° = e Recall, from Tt x ) ( x ) [2.4»1]» s ( > that e » x arbitrary} . the double morphism vE : TE = x X TM M is E an E-morphism having an E - k e r n e l V . U e U vf e = In l o c a l coordinates, (x,u,dx,du)| 'vj points o = { ( x , u ,0,du) that that T^E. e o f TE a r e o f t h e form and, fpr e = ( x , u ) , S o , we must e s t a b l i s h We c l a i m U*e 0 vf i s given by: du € E } . k contains a l l points o f the 53 form ( x , u ,dx,du) with dx ^ Oo Note that, f o r u and k n du i n IR , and dx / 0 i n K , we can f i n d a smooth map 0 0 0 F : TJ >Ek, where TJ i s some neighbourhood of x , such that F(x ) 0 = u, and F'(x )dx 0 = du. Now, there e x i s t s a vector f i e l d X on M with X(x ) = dx, e and a section S o f E which agrees with F on a neighbourhood of x , so we are done. 0 APPLICATION; I f K and K are connection forms on E rV Inducing the same covariant d e r i v a t i v e , then K = K. PROOF; By d e f i n i t i o n , K and K agree on V , and by E hypothesis they agree on each U as w e l l . Therefore, by our e a r l i e r claim, the r e s u l t . I 3.2.2; Existence Let M be a smooth manifold of dimension n, and l e t p :E >M denote a smooth vector bundle on M with fibre E . k Let V be a covariant derivative on sections of E. We wish to define a connection form on E inducing V as i n [ 3 - 1 ] . By [ 3 . 2 . 1 ] i t s u f f i c e s to f i n d K l o c a l l y . Recall that i n l o c a l coordinates we may w r i t e : VS X s xi *(x,u'(x) c5 (x) + T (u(x), t| (x))) x 54 where -f : IR X k Ht IR i s b i l i n e a r and n k depends smoothly on x. We define K l o c a l l y by K s (x,u,dx,du) (x,du + P(u..dx) ). I t i s e a s i l y v e r i f i e d that the global -morphism K. so. defined s a t i s f i e d the conditions of a connection form. Section 3.3: Differences Note, i n the foregoing notation, that i f K and K* are connection forms on E inducing covariant d e r i v a t i v e s V and V r e s p e c t i v e l y , then the difference f i e l d defined i n [2.6.2], and the difference tensor D ( V , V ) of [1.3.1] are i d e n t i c a l a f t e r the usual i d e n t i f i c a t i o n of operators V(M) X r ( E ) K —> rjj(E) with smooth morphisms TM X M E which are l i n e a r fibrewise. > E 55 Chapter 4? Sprays S e c t i o n 4ols § 4.1.1? Preliminaries The Double Tangent Bundle Let M be a smooth manifold of dimension n. Denote the tangent bundle of -M" by •tfa s TM : —* M and r e c a l l that i t i s a smooth vector bundle over M with f i b r e 3R . Since TM i s a smooth manifold of n dimension n , i t has a tangent bundle, c a l l e d the double 2 tangent bundle of M, which we denote by TTfyj : TTM —-. > TM. Recall that i f a point of M l i e s within the domain of a chart (U, <j?), then we denote i t by i t s image x<= f ( U ) i n m . Also, i f (U, <f> ) and ( V , n are charts having a non-empty i n t e r s e c t i o n , W, then there i s a coordinate change on W given by a diffeomorphism h : <p(W) ——» *K )» W At the tangent l e v e l , points of the corresponding domain are denoted by pairs (x,dx) e <^(U) X 3R , and the con ordinate change i s given by the diffeomorphism <P(W) x m (x,dx) 1 n —> ; VL(W) x m n > (h(x),h'(x)dx). Since TTM i s the tangent bundle of TM, we w i l l 56 denote points by quadruples (x,dx,x,dx) € (y(U)><dR ^dR ><3E , r n n and the corresponding coordinate change here i s given by CD(W) X ffi ( x o d X y X o d x ) 1 4,1,2: X l n n X f f i > ^(W) n X E N X JR N X E N > (h(x),h (x)dx hUx)x,h"(x)xdx+hUx)dx) v t 0 The Double Structure of TTM Let M be a smooth manifold of dimension n. * Tfy TM Denote by >M the tangent bundle of M, and by Uj^ ' TTM the double tangent bundle. » TM Since TM i s a vector bundle over M, TTM w i l l have the double structure discussed i n [2.2], In p a r t i c u l a r , n : m TTM > TM gives the standard structure of TTM, and we have the following exact sequence of smooth vector bundles over TMs i <ft 0 y TM X TM TTM > TM X TM > 0, M M where, i n l o c a l coordinates, ttq^j : (x,dx,x,dx) H i Ti > (x,dx), : (x,dx,x)* »(x,dx,0,x), J (x,dx,x,dx)i > (x,dx,x). and On the other hand, the tangent structure of TTM i s given by: TT^ : TTM >TM, and another exact sequence over TM i s given by: 57 •4 TM X -> TTM TM M -> TM X TM M » 0, where, again i n l o c a l coordinates, Tr^j c (x,dx,x,dx) i > (x,x), and j : (x,dx,x)i >(x,0,x,dx). I t i s important to note that these two structures are That i s , I f I ^ r ^ denotes the i n v o l u t i o n of Isomorphic TTM given l o c a l l y by s 'TTM and I TM T M (x,dx,x,dx)i *(x,x,dx,dx), denotes the canonical i n v o l u t i o n of TM X M TM, then the f o l l o w i n g diagram commutes: -* TM X M i TM -•TTM I TTM •TM -> TM X M TM Section 4.2: I 4.2,1: —> TM X M •> TTM I •^TM X M TM ->0 TM TM -»0. Bundle Connections on Manifolds Definition Let M be a smooth manifold of dimension n, with tangent bundle : TM > M. We define a connection map on M to be a connection map on the vector bundle TM as i n [2o5'.l]. C of Tf That i s , a smooth section : TM X M TM » TTM (as defined i n [4<>1»2]) which i s a vector bundle morphism with respect to both the standard and tangent 58 structures of TTM. S i m i l a r l y , a connection form on M i s defined to be a connection form on TM. That i s , a smooth morphism K s TTM. —-. TM satisfying% CF 1 s K i s a vector bundle morphism whose r e s t r i c t i o n to i(TM X M TM) i s projection onto the second coordinate, and CF 2 : K i s a vector bundle morphism over whose r e s t r i c t i o n to the where i j(TMX M TM) i s also p r o j e c t i o n onto second coordinate, and j In [ 2 , 5 , 3 ] , are as i n [4e>1.2], we saw t h a t c o n d i t i o n CF 2 i s equivalent to CF 3 where K s e IR o s^ = s K denotes fibrewise m u l t i p l i c a t i o n by s: y TM s 1 TM (x,dx)i and o *(x,sdx), s^. the tangent map 0 § 4.2.2; Connection Maps versus Connection Forms In [3»5] i t was shown that connection maps and connect i o n forms are i n a one-to-one correspondence. R e c a l l that l o c a l l y , connection maps have the form 59 ! C s (x,dx,x)i > (xydXyXp-^CdXjx) ) where Q i s a b i l i n e a r mapping of JR X lR onto B n depending smoothly on x. n n Corresponding to each such C we have a connection form K^, given l o c a l l y by KQ S ( x j d X y X p d x ) t (x,dx + P^C dx,x) ). Since the two bundle structures of TTM are isomorphic, we have a notion of symmetry. That i s , we say that a connection map C on M i s symmetric i f i t commutes with the i n v o l u t i o n s of TTM and TM X TM introduced i n [4.1-2]. That i s , when "'° % c = I TTM e C o Correspondingly, a connection form K on M i s symmetric when K ° TM T = * K In l o c a l coordinates, t h i s means K and C are symmetric i f and only i f each of the corresponding P *s i s symmetric. Thus C i s symmetric i f and only i f i s symmetric. F i n a l l y , r e c a l l that f o r any two connection maps C and C on M there i s a morphism D(C,C) : TM X TM >TM M c a l l e d the difference f i e l d of C and C, and that given any 60 connection map C, and a morphism D as stated,, there i s a unique connection map C on M such that D(C,C) = D. No.te that i f C and C are symmetric, then so- i s DtC,C). Section 4.3: § 4.3.1: Second Order D i f f e r e n t i a l Equations and Sprays Second Order D i f f e r e n t i a l Equations Let M be a smooth manifold of dimension n, s TM * M i t s tangent bundle, and T T ^ S TTM >TM i t s double tangent bundle (with standard s t r u c t u r e ) . Recall that Tn^ : TTM TM also gives TTM a smooth vector bundle structure (the tangent structure) and that the canonical i n v o l u t i o n I : TTM $> TTM i s an isomorphism of these two structures. A second order d i f f e r e n t i a l equation on M i s defined to be a smooth vector f i e l d X : TM » TTM which i s a s e c t i o n of both the standard and tangent structures of TTM. That i s , such that TM or equivalently, = n o X I X 0 id TM ° T n M° ' X as X . In l o c a l coordinates, then, X must have the form X : (x,dx) i > (x,dx,dx, ^ (x,dx)) 61 where ^ i s a smooth IR ~valued n By d e f i n i t i o n , p i s an i n t e g r a l map. a smooth curve : I > TM curve of a vector f i e l d Y on TM i f and only i f p'(t) where p ' = ; Y(^(t)) I f o r each t i n I, > TTM i s the canonical l i f t of /3 to the tangent bundle of In l o c a l coordinates, p may be written > (cr ( t ) , t ( t ) ) ti TM. € U X B ^ Therefore, i f X i s a second order d i f f e r e n t i a l equation with l o c a l representation (x,dx)i p i s an i n t e g r a l » (x,dx,dx, ^(x,dx)), curve of X i f and only i f s cr'U) = t(t), t'(t) = <f ( c r ( t ) , t ( t ) ) and In other words, the condition i s that cr " «|(cr,cr'). X = CT', and Therefore, i n t e g r a l curves of second order d i f f e r e n t i a l equations are equal to the canonical l i f t s of t h e i r projections onto M. I 4o3.2s Sprays We continue with the notation of the l a s t Let C : TM >< TM M > TTM subsection. 62 be a connection map on M as defined i n [4«2.2], I f we r e s t r i c t C to the diagonal of TM X TM, then M the r e s u l t i n g map induces a second order d i f f e r e n t i a l equation . X TM % Q c a l l e d the spray of C. —> TTM In l o c a l coordinates, i f C i s given by C : (x,dx,x)i then »(x,dx,x,- (dx,x)) i s given by XQ : (x,dx)i > (xjdx^dXy-r^CdXydx)). We may characterize those second order d i f f e r e n t i a l equatxons which a r i s e from ccnnectxon maps as f e l l o w s ! For s € R, l e t and s : TM > TM s s TTM » TTM denote f i b r e w i s e m u l t i p l i c a t i o n by s, and l e t s^. denote the tangent map of s Then i f X^ i s the spray of a 0 connection map C on M, the f o l l o w i n g condition holds: (Sp) s*<> s * X since each Hy. i s quadratic. C = X ° s, C I n general, a smooth morphism X : TM * TTM i s c a l l e d a spray on M whenever i t i s a second order d i f f e r e n t i a l equation on M s a t i s f y i n g condition (Sp). L o c a l l y then, i f X i s given by X % (x,dx)l > (x,dx,dx, <S (x,dx)), 63 condition (Sp) implies <^(x,sdx) = s ^(x,dx), 2 so a second order d i f f e r e n t i a l equation i s a spray i f and only i f | i s a quadratic form i n dx. Section 4.4 s The Connection Map of a Spray Let M be a smooth manifold of dimension n, and l e t X s TM — » TTM be a spray on M as defined i n the l a s t section. THEOREM; There i s a unique symmetric connection map C ; TM X TM M on M generating the spray X. PROOF; > TTM Suppose f i r s t that C and C are two symmetric connection maps generating the same spray X. t h e i r d i f f e r e n c e as i n [ 2 . 6 . 1 ] . Consider By hypothesis, i t w i l l vanish on the diagonal of TM X TM, and so must be a l t e r M nating. On the other hand, i t i s symmetric. Therefore, A/ i t must vanish everywhere. Thus C = C. Since uniqueness i s established, we need only exhibit a suitable map C l o c a l l y . X : (x,dx)t where jpx(dx) Suppose X i s given l o c a l l y by > (x,dx,dx, <| (dx)) x «= ^(x,dx) i s quadratic i n dx. Since^ x i s homogeneous of degree two, i t i s the r e s t r i c t i o n to the diagonal of some b i l i n e a r mapping of 3R X n (cf Appendix). IR into lR n The family of a l l such maps inducing | x n 64 contains a unique symmetric member. by -To Denote t h i s map The following i s then a well-defined connection map: C : TM X TM > TTM M (x,dx,x) i > (x,dx,x,- (dx,x)) and s a t i s f i e s the hypothesis of the proposition. As a consequence of t h i s theorem, note that given any connection map connection map spray. C on M, there e x i s t s a unique symmetric S S C on M such that C and C induce the same Setting, f o r an a r b i t r a r y connection map T(C) = D(C,C ) : S TM X TM C on M, * TM, M we obtain an alternating tensor f i e l d T, c a l l e d the t o r s i o n f i e l d of C. Thus, although sprays are not i n one-to-one correspondence with bundle connections or covariant derivatives, the e x p l i c i t formulas i n l o c a l coordinates do imply the following r e l a t i o n s h i p . Let V be a covariant derivative on M, l e t K be the corresponding connection form, and C the corresponding -connection map, and l e t X be the corresponding spray. Then: (1) V (2) a smooth curve on M i s a geodesic of V i f and only i f i s torsion-free i f and only i f K and C are symmetric, i t i s the projection to M of an i n t e g r a l curve of X, and (3) the t o r s i o n tensor of V is equal to the t o r s i o n f i e l d of C. 65 BIBLIOGRAPHY 1. W. AMBROSE, R. S„ PALAIS, I, M. SINGER^ "Sprays", Anais du Academia B r a s i l e i r a de Ciencias, v o l . 32, 163-173 ( I 9 6 0 ) 2 J . DIEUDONNE: "Elements d'Analyse", tome I I I , Gauthier-Villars, Paris, (1970) E 3. Co GODBILLONs "Geometric D i f f e r e n t i e l l e et Mecanique Analytique", Hermann, Paris, ( 1 9 6 9 ) 4. D. GROMOLL, W. KLINGENBERG, W. MEYERs "Riemannsche Geometrie im Grossen", Springer Lecture Notes, Berlin-Heidelberg-New York, 0-968) 5. N. Jo HICKS: " D i f f e r e n t i a l Geometry", Van Nostrand, Toronto-New York-London, (1965) 6. S. LANG: "Introduction to D i f f e r e n t i a b l e Manifolds", J* Wiley; New York-London (1962) ; 66 APPENDIX Homogeneous Smooth Functions Recall that a function f s & > B* n 1 i s homogeneous o f degree k ^ 0, i f and only i f , f o r each u €B , (H) n f(su) f o r any s e B . s f(u) k Note that i f f degree 0, then (H) implies that function, and i f f i s homogeneous of f i s a constant i s homogeneous of degree k > 0, then f(0) = 0 . LEMMA: If f i s a d i f f e r e n t i a b l e homogeneous f u n c t i o n of degree k > 0, then Df : B > Lin(B , B ) n n 01 i s homogeneous of degree k - 1 . PROOF: D i f f e r e n t i a t i n g (H) with respect to u gives; sDfl = s Df| , lu DfI l(su) = s l(su) k so LEMMA: (Euler's Relation) k _ 1 DfI l(u) If f i s a homogeneous d i f f e r e n t i a b l e function of degree k > 0, then Df| (u) lu PROOF: = kf(u). D i f f e r e n t i a t i n g (H) with respect t o s Df I l(su) (u) = ks^fCu). gives But by the above lemma. Df| (su) (u) o s ^ D~f (u), u t l so we are done. As a consequence of these two lemmas, we have the following proposition which may be proved by induction on k, P R O P O S I T I O N : F S A smooth function ^m J R H i s homogeneous of degree k > 0 i f and only i f there e x i s t s a k - l i n e a r function f : IR X n B n X ..... X ]R such that f o r each u € IR , 11 £(u,u, .....,u) =s f(u). n ?TR m
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Connections Nicolson, Robert Alexander 1972
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Title | Connections |
Creator |
Nicolson, Robert Alexander |
Publisher | University of British Columbia |
Date Issued | 1972 |
Description | The main purpose of this exposition is to explore the relations between the notions of covariant derivative, connection, and spray. We begin by introducing the basic definitions and then use a method of Gromoll, Klingenberg and Meyer to show that covariant derivatives and connections on vector bundles are in a natural one-to-one correspondence. We conclude by showing that on the tangent bundle of a manifold, sprays and "symmetric" connections are in a natural one-to-one correspondence. Although we use a different method, this re-establishes a result of Ambrose, Palais, and Singer. |
Subject |
Connections (Mathematics). |
Genre |
Thesis/Dissertation |
Type |
Text |
Language | eng |
Date Available | 2011-04-13 |
Provider | Vancouver : University of British Columbia Library |
Rights | For non-commercial purposes only, such as research, private study and education. Additional conditions apply, see Terms of Use https://open.library.ubc.ca/terms_of_use. |
DOI | 10.14288/1.0080458 |
URI | http://hdl.handle.net/2429/33610 |
Degree |
Master of Science - MSc |
Program |
Mathematics |
Affiliation |
Science, Faculty of Mathematics, Department of |
Degree Grantor | University of British Columbia |
Campus |
UBCV |
Scholarly Level | Graduate |
Aggregated Source Repository | DSpace |
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