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UBC Theses and Dissertations

Connections Nicolson, Robert Alexander 1972

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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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