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UBC Theses and Dissertations

Lie transformation groups Thomson, James P. 1974

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LIE TRANSFORMATION GROUPS by JAMES P. THOMSON B.Sc., University of Bri t i s h Columbia, 1972 A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF SCIENCE in the Department of MATHEMATICS We accept this thesis as conforming to the required standard THE UNIVERSITY OF BRITISH COLUMBIA October, 1974 In presenting th is thes is in p a r t i a l fu l f i lment of the requirements for an advanced degree at the Un ivers i ty of B r i t i s h Columbia, I agree that the L ibrary sha l l make it f ree ly ava i l ab le for reference and study. I fur ther agree that permission for extensive copying of th is thes is for scho la r ly purposes may be granted by the Head of my Department or by his representat ives . It is understood that copying or pub l i ca t ion of th is thes is for f inanc ia l gain sha l l not be allowed without my wri t ten permission. Department of MdHi-e m4T~ IC £ The Univers i ty of B r i t i s h Columbia Vancouver 8, Canada Date NflV/. 25: / 7 4 - i i -ABSTRACT Suppose G i s a Lie group and M i s a manifold (G and M are not necessarily f i n i t e dimensional). Let D(M) denote the group of diffeomorphisms on M and V(M) denote the Lie algebra of vector fields on M . If X: isv a. complete-vector: field- then; Exp tX w i l l denote the one-parameter group of X . A local action <£ of G oh M gives rise to a Lie algebra homomorphism <J>+ from L(G) into V(M) . In particular i f G i s a subgroup- of D(M> and <|> : G x M —> M i s the natural global action (g»p) —> g(p). then G i s called a Lie transformation group of M. If M i s a Hausdorff manifold and G is a Lie transformation group of M we show that <j> i s an isomorphism of L(G) onto <f> (L(G)) and L = <j>+(L(G)) satisfies the following conditions : (A) L consists of complete vector f i e l d s . (B) L has a Banach Lie algebra structure satisfying the following two conditions : (BI) the evaluation map ev : (X,p) —> X(p) is a vector bundle morphism from the t r i v i a l bundle L x M into T(M), (B2) there exists an open b a l l B r(0) of radius r at 0 such that Exp : L — > D(M) i s infective on B r(0). Conversely, i f L i s a suba-lgebra of V(M) (M Hausdorff) satisfying conditions (A) and (B) we show there exists a unique connected Lie transfor-+ mation group with natural action <j> : G x M —> M such that tj) is a Banach Lie algebra isomorphism of L(G) onto L . Table of Contents Chapter 1 Preliminaries §1 Foliations, and Integrah-le*Subbundlest §2 Total Differential Equations §3 Lie Groups and Lie Algebras-Chapter 2 Local and Infinitesimal Group Actions §4 Local Group Actions §5 Infinitesimal Actions §6 The Infinitesimal Graph §7 Existence Theorem §8 Uniform Infinitesimal Actions Chapter 3 Connected Lie Transformation Groups §9 The Image of the Infinitesimal Generator of a Lie Transformation Group §10 Banach Lie Algebras of Complete Vector Fields §11 A Banach Lie Algebra of Complete Vector Fields which does not generate a Connected Lie Transformation Group Bibliography - i v -Acknowledgements I wish to thank Dr. Ottmar Loos for the suggestion of this thesis topic and the many helpful suggestions arid criticisms throughout i t s development. I would also lik e to acknowledge Dr. Jens Gamst who i n i t i a l l y generated my enthusiasm,,for-, differential-, geometry.. Appreciation i s expressed to the Department of Mathematics for providing financial support during the period of my graduate studies at the University of British Columbia. - 1 -Chapter 1 Preliminaries A l l manifolds considered are real Banach. manifolds of class C where K = «° or K = to . The word morphism w i l l mean a C map between K C manifolds. In this chapter, we collect the necessary facts on foliations of manifolds and on i n f i n i t e dimensional Lie groups. Almost a l l of this material w i l l come from Bourbaki [1, §9] or Bourbaki [2, Chapter 3], . _ §1 Foliations and Integrable Subbundles Let M and S be manifolds and p : M —> S a submersion. We then have, for each s e S, a manifold structure induced on the level set p ''"(s) by M . Denote by the manifold which i s the disjoint union over S of p ^(s ) . Each p ^"(s) i s an open submanifold of and topologically i s the topological sum of the topological spaces p ^"(s). Definition (1.1) Let M be a manifold. A foliation of M is a manifold Y having the same point set as M. and satisfying the condition that for a l l x e M, there exists an open submanifold U of M containing x, a manifold Sy and a submersion p : U —> S such that the manifold i s an open submanifold of Y.. The inclusion map of Y into M i s easily seen to be a bijective - 2 -immersion. We c a l l the pair (M, Y) a foliated manifold. A set U is called a (connected) leaf i f i t i s -a (connected) open set i n Y . The maximal connected leaves are therefore the connected components of Y. Definition (1.2) If (M, Y) and .(M1, Y') are foliated manifolds, a morph ism from (M, Y) into (M' , Y'1) is a map which is a morphism of M into M' and. at the same time a morphism of Y into Y' . Using the inclusion map of Y into M, for each x e M we can identify the tangent space T x(Y) with a subspace of T x(M). With this identification we have the following propositions. Proposition (1.3) The spaces T X(Y) are the fibers of a subbundle T(M, Y) of T(M). Furthermore i f Y i s defined by a submersion p : M —> S , then T(M, Y) = ker T(p) . Proposition (1.4) Let (M, Y) and (M*, Y') be two foliated manifolds and f : M —> M' be a morphism. A necessary and sufficient condition that f  i s a morphism from (M, Y) into (M', Y') i s that T(f) takes T(M, Y) into T(M*, Y'). Let. F be a subbundle of T(M). We now examine the conditions on F which imply the existence'of a manifold Y such that T(M, Y) = F . If this i s the case then F i s called an integrable subbundle-of T(M) and the foliation i t defines i s unique. - 3 Theorem of Frobenius (1.5) F is integrable i f there exists a family {£.}.,. of sections of F such that 1 l e i —: : (1) for a l l x e M .the set {£^(x) : i e 1} is a total subset of the fiber F of F above x . — . x — (2) for a l l pairs ( i , j) of elements of I and a l l x e M , [ q . ^ ( x ) e F x . §2 Total Differential Equations We now construct a particular subbundle and examine what i t means for i t to be integrable-. This w i l l be the setting for discussing 'generalized di f f e r e n t i a l equations. Suppose M is the product of two manifolds A and B. Let p^ : M — > A and p 2 : M —> B be the projections on the f i r s t and second factors. There are two subbundles, p 1*T(A) and p 2*T(B), of T(M) = T(A) x T(B) associated with p 1 and p 2 . The fiber P j * T ( A ) ^ b^ of p 1*T(A) over (a,b) i s T a(A) x {0b> where 0 b is the zero vector i n T b(B). We identify this fiber with T a(A). Similarly the fiber P 2*T(B) ( a b ) of p 2*T(B): over (a,b) i s {0fl} x T b(B) which i s identified with T b(B) . Let f be a vector bundle morphism from p^*T(A) into p 2*T(B). Then for each (a,b) e M, f is a continuous linear map f(a,b) : T a ( A ) _ > T b ( B ) (after identifying T Q(A) with p 1 * T ( A ) ( a b ) _, 4 -and Tb(B) with P2*T(B) b ) ). Proposition (2.1) The graphs of the f ^ ^  are the fibers of a subbundle  of T(M) which we denote-by- F^ . Definition (2.2) Let A' be an open set in, A . A morphism $ : A' —> B is,called an integral of f- if- for a l l a. e A'- one has T (*) = f, ,, SN. 52 a (a,cf>(a)) The following two propositions describe the local uniqueness of integrals. Proposition (2.3) If (f>^  and <j>2 are two integrals of f taking the  same value at a point a e A, then they coincide in a neighbourhood of a. Proposition (2.4) Let Z be a manifold, A1 an open set in A, and a e A' . Suppose cfi^ and §^ are morphisms of Z * A' into B such that <j)^  and coincide on Z x {a} and for a l l . z e Z, the morphisms a —> <j>^ (z,a)' and a —> <j>2(z,a) are integrals of f . Then cfi^ and  coincide on a neighbourhood of Z x {a} . Suppose now that F^ i s integrable and therefore defines a foliation Y of M with T(M, Y) = F f . Let o> : A' —> B be an integral for f. and define ty : .A' ••—> M by i|»..(,a) = (a, <j>,(a)).. .We have Tip(T(A')) C F^ since .<j> was an integral and Proposition (1.4) gives that ty i s also a morphism from A' into Y . Let v e T A1 . Now a a W " ( va» V'V* = ( v a ' f ( a . ( a ) ) ^ ^ w h i c h i m p l i e s T a * i s a n - 5- -isomorphism (of Banach spaces) of T A' onto F^. ,, • This means ty 1 ' a (a,<j>(a)) i s a local diffeomorphism into Y at a and as a was arbitrary we have proven the following result. Proposition (2.5) If_ F^ is --int e grab le and- cj> : A' —> M i s an. integral  for f then {(a, cj>(a))« :. a e A' } is a. leaf (open set) of the foliation  defined by F^ . We complete this section with the existence theorem for integrals. Proposition (2.6) Suppose that F^ is integrable. Let (Z q, a Q) e Z x A and p be a morphisny from- Z» into- B- Then there exists- an- open  neighbourhood Z' x A' of. ( Z Q J a Q) in Z x A and a morphism <J> : Z' x A' > B such that for every z e Z1 the morphism a —> cj)(z,a) of A' into B 1 i s an integral for f and p(z) = <}>(z,ao) . We w i l l mainly use this with Z = B and p = identity. §3 Lie Groups and Lie Algebras A Lie group G i s a group, which is also a Banach manifold (not necessarily f i n i t e dimensional) such that the operations of multiplica-tion G x G —> G and taking inverses G —> G are morphisms. G w i l l be called f i n i t e (infinite) dimensional i f i t s manifold structure i s modelled on a f i n i t e (infinite) dimensional Banach space. '. - 6 -A Banach Lie Algebra L is a Lie algebra with a Banach .space structure such that the bracket [ , ] : L x L —> L i s continuous. We c a l l L f i n i t e (infinite) dimensional i f the underlying vector space is f i n i t e (infinite) dimensional. Almost a l l of the standard f i n i t e dimensional Lie group theory carries over to i n f i n i t e dimensions. If G i s a Lie group then there is a Banach Lie algebra L(G) corresponding to G arid an exponential map from L(G) into G, which is a local diffeomorphism at 0 . (We break with the usual convention of having L(G) equal to the set of l e f t invariant vector fields on G and instead i t w i l l be the set of right invariant vector fields. Defining L(G) to be the right invariant vector fields w i l l make the definition of an infinitesimal action in Chapter 2 easier. This i s a slight change since i f we identify L(G) with T e(G), the tangent space at the identity, then the only difference between the right invariant Lie algebra structure and l e f t invariant Lie algebra struc-ture i s that the bracket differs by a sign.) The major difference between the f i n i t e and i n f i n i t e dimensional theories is that there exist i n f i n i t e dimensional Banach Lie algebras L for which there does not exist any Lie group G such that L = L(G). If a Lie group G does exist such that L = L(G) then the Banach Lie algebra L i s called enlargeable. For an example of a non-enlargeable Banach Lie algebra see Est and Korthagen [4]. Although L may not be enlargeable, a Banach Lie algebra closely related to L is always enlargeable. This Lie algebra i s the path space of L which we now examine. - 7 -Let BL denote the category of Banach Lie algebras with conti-nuous homomorphisms as morphisms. Then we have the path functor A : BL —> BL which takes L to AL = {f | f : [0,1] —-> L continuous with f(0) = 0 } with the following Lie algebra structure. If f, g e AL then the norm of f i s max | | f ( t ) | | and the bracket i s defined te[0,l] pointwise, [f, g] (t) = [ f ( t ) , g(t)]. If cf> : L —> L' i s a morphism of Banach Lie algebras then Act : AL —> AL' is given by Act'(f) = <f>°f . Theorem (3.1) Let L be a Lie algebra and AL be as above. Then (1) the -endpoint evaluation map f —> f ( l ) from AL into L is continuous. (2) AL is enlargeable. Proof : The proof of (1) is obvious from the definition of AL . The reader i s referred to Swierczkowski [8] for a proof of (2). For later reference, we now l i s t some facts on subgroups and subalgebras of Lie groups and Banach Lie algebras. The proofs are in Bourbaki [2, Chapter 3]. Definition (3.2) A subset H of G i s a Lie subgroup of G i f i t i s a subgroup and a submanifold of G . Proposition (3.3) Let H be a subgroup of a Lie group G . A necessary and sufficient condition for H to be a Lie subgroup is that there exists  a point h e H and an open neighbourhood U of h in G such that - 8, -H fl U i s a submanifold of G .. Let L be a Banach Lie algebra. A Banach Lie subalgebra of L i s a closed vector subspace of L which is closed under, the bracket operation, i.e. a subalgebra. If H is a Lie subgroup of a Lie group G then using the inclusion we identify L(H) with a Banach Lie subalgebra of L(G) which sp l i t s i n L(G). (A closed subspace F of a Banach space E i s said to s p l i t i f there exists a closed subspace F^ such that F + F 1 = E and F f ^ = 0 ). If i n addition H i s normal then L(H) i s an ideal in L(G), i.e. [L(G), L(H)] C L(H). Proposition (3.4) Let G b,e a,Lie., group and H be a no.rmal Lie subgroup of G . Then there exists.a structure of a Lie group on G/H such that  the projection map i s a submersion and L(G/H) = L(G)/L(H) . Proof : Bourbaki [2, prop. 11, p.105 and p.141] . - 9" -Chapter 2 Local and Infinitesimal Group Actions We determine the correspondence between local group actions and infinitesimal group actions in "this chapter. Our treatment of this subject follows that of Palais [7]. Before proceeding we establish some notation conventions. G w i l l denote a connected Lie group and L(G) w i l l be i t s Banach Lie algebra of right invariants vector fields. Right multiplication, by an element g e G w i l l be denoted by R(g) . The Identity element in. G w i l l be denoted by e . M w i l l denote a manifold and V(M) w i l l be the Lie algebra of vector fields on M . §4 Local Group Actions Definition (4.1) A local (left), action of G on M is a morphism <j> from an open set D containing {e} x M in G x M into M satisfying the following conditions : (1) <|>.(e, p) - p for a l l p e M . (2) If (h, p), (g, <Kh, p)) and (gh, p) a l l belong to D then <j>(gh, p) = Kg. <f>(h, p)) • If D = G x M then <{> is called a global action of G on M . - 10. -Let D P = { g : (g, p) e D } . The morphism g —> <j>(g, p) of D P into M w i l l be denoted by <J>P . The definition of local action we have given i s from Palais [7]. Bourbaki [2, p.118] gives what appears to be a different definition of local action as follows. Definition (4.1(a)) (Bourbaki) A local (left) action of G on M i s a morphism ty defined on an open set ft of G x M containing {e} x. M , with values i n M, possessing the following properties (1) ty(e, p) = p for a l l p e M ; (2) there exists a neighbourhood ft^ of .{e} x .{e} x M i n G x G x M such that, for (g, g', p) c ft^, the elements (g 1, p), Cgg'> P)> (g. Mg', P)) are in ft and ty(g, ty(g', p)) = ty(gg', p) . This i s slightly different from the version in Bourbaki since we aren't considering actions of "grouplets". Proposition (4.2) Definition (4.1) and Definition (4.1(a)) are equivalent. Proof : Def.(4.1) implies Def.(4.1(a)) Let ty = <j) and ft = D . We have to find an open set ft^ i n G x G x M satisfying condition (2) i n Def.(4.1(a)). Define 6 from G x D into G x M by 6(g,. h, p) = (g, <j)(h, p)), then 5 ^(D) is open and contains {e} x {e} x M . Define y from G x D into G x M by Y(g» h, p) = (gh, p) then y *(D) i s open and contains {e} x {e} x M . - 11 -Let ft^ = 6 "'"(D) H y "^(D), then ft^ i s an open neighbourhood of {e} x {e} x M and i f (g, h, p) e we have (h, p) e D; (g, (j)(h, p)) e D since (g, h, p) e 6 "''(D); and (gh,, ,p) e D since (g, b, p) e y ^ " ( n ) « Then Def.(4.1) (2) gives Kg,ty(h, p)) = Mg, $(h, p » = eV(gh, p) = \Hgh, p) and condition (2) of Def.(4.1(a)) i s satisfied. Def.(4.1(a)) Implies Def.(4.1) Let <f> = ty . We w i l l find D such that condition (2) of Def. (4.1) i s s a t i s f i e d . Let ft^ be as i n Def.(4.1(a)). Since ft^ is an open neighbourhood of {e} x {e} x M we can find neighbourhoods and of e i n G and W of p in M such that V x TJ x w C I ft, . For P P P P 1 each p e M, l e t G = exp (15^(0)) where 15^ (0) i s the b a l l of radius r centered at 0 in L(G) and r is so small that G C v H U • Then P P P G i s connected, G = G ~ 1, G C U , G C V , and G x G x W i s P P P P P P P P P P an open neighbourhood of (e, e, p) contained in ft^ . Also ^p^p e^ a r e ordered by inclusion so i f we have G and G then either G C G or x y x y G C G • Define D = ( J G x w and suppose (h, p), (g, <j>(h, p)) and 7 X peM P P (gh, p) e D . Since D i s "symmetric" (each G^ was symmetric) we have (h, p) e D implies (h \ p) e D . Now by the definition of D ; (gh, p) and (h p) belonging to .D means there exists x e M such that (h 1 , p) e G x W and there exists y z M such that (gh, p) e G x w '. x x ' & ' v y y By the remark above either G C G or G (~ G so (without loss of x y y x generality) assuming the latter we have (gh, p) e G x W also. Now X X Gl C V n U implies ((gh)(h - 1), p) e G\ x w. . C V y x W , i.e. A A. A. - A A A A (g, p) e V x .w . We also have (h, p) e G x W CT U x W which means X X X X X X (g, h, p) e V x U x W G fi, and condition (2). of Def. (4.1(a)) gives X X X X tHg", <K-h-> P')'> K"g-» vb-(-h-, p)-)* =*Hgh'» p ) = ( f i'(gh, p). Examples of local actions Example (4.3) : Let H be a paracompact manifold and £ be a vector f i e l d on M . Then the flow (see Bourbaki [1, §9]) of 5 i s a local l e f t action of CR. on M . Example (4.4) : If E and F are Banach spaces then denote by Hom(E, F) the Banach space of continuous linear maps from E into F and by GL(F) the Lie group of invertible elements in Hom(F, F). GL(F) i s open in Hom(F, F). (See Lang [5, p.5] for proofs). Let M = Hom(F, E), G be the additive Lie group Hom(E, F), and I„ be the identity in GL(F). Define r the morphism y : G x M —> Hom(F, F) by y(g> P) = g°P + 1- • Let r D = y 1(GL(F)); then D i s open and contains {0} x M. Define the local action <j> : D —> M of G on M by c{>(g, p) = P°(g°P + I - p ) ^ • <(> i s a r local action for ; CD cK-o-, p) = p°(p + i F ) _ 1 = P (2) <f>(g, ;<Kh,. PO). = <Kh,• p)o(g.o*(h,,.P). + i F ) _ J -= p°(hop + I p) 1(g 0p 0 (hop + i p ) 1 + I p) 1 = po((gop0(ho-p + I F ) _ 1 +• I F)(hop + I F ) ) - 1 - p ° ( g ° p + hop + r F ) 1 = P°((g + h ) o p + i p ) 1 = <{)(g + h, p) . §5 Infinitesimal Actions Let L be a Banach Lie algebra. Definition (5.1) A (left) action of L on M i s a Lie algebra homomorphism 6 : L —> V(M) satisfying the condition that the evaluation map (x, p) —> 8(x.)(p) i s a vector bundle morphism from the t r i v i a l vector bundle L x M into T(M). Remarks : (1) If L = L(G) for some Lie group G then 0 is called an infinitesimal (left) action of G on M . (2) If L is f i n i t e dimensional then the evaluation map is automatically a vector bundle morphism (Bourbaki [2, Remarque p.140]). - 14 -Example (5.2) : An infinitesimal group action Suppose H i s a real Hilbert space with scalar product ( , ) . Let M = H and G be H with the additive group structure of H . Then L(G) = H also. Define 6 : L(G) —> V(M) by 6(Y)(X) = 2(X,Y)X - (X,X)Y. We show that 0 i s an infinitesimal action of G on M . (1) The map e : (Y,X) —> 6(Y)(X) i s a vector bundle morphism from L(G) x M into T(M) : e i s obviously a morphism. Let Hom(H, H) denote the continuous linear maps from H into H and let ^ e H'om(H, H) be the map Y — > G (Y) (X). We need that the map X —> 8 of H into x Hom(H, H) is continuous, but this i s the case since ( , ) : H x H —> H is continuous. (2) 0 i s a Lie algebra homomorphism : 0 is obviously linear. In order to prove that 0 preserves brackets i t suffices to show that [0(Y), 0(Z>] = 0 for any Y and Z in L(G) since L(G) = H is abelian. By definition [0(Y), 0(Z)](X) = D0(Z)| y(0(Y)(X)) - D0(Y)| x(0(Z)(X)) . A short calculation gives D9(W)| (H) =2 (X,Y)H + (H,Y)X - (X,H) and substituting this into the above equation with W = Z (and Y) and H = 0(Y)(X) (and 0(Z)(X) ) makes the equation identically zero. Therefore 0 preserves brackets. Suppose <j> : D —> M is a local action of G on M . Define V"": L (G) —> V(M) by <}>+(v) (p) = T(<j>)(v(e), 0 ) where 0 p is the zero vector in T (M) . P Proposition (5.3) cj>+ is. an infinitesimal action of G on M . + Proof : Evaluation map of <j> is a vector bundle morphism : We have the following sequence df maps L (G) x M —> L (G) x G x. M — ^ % . T D — > TM (v, p) > (v, e, p) > (v(e), 0 ) > T(<^)(v(e), 0 ) where *£ i s the t r i v i a l i z i n g vector bundle isomorphism (v, g) —> v(g) of L (G) x G into T(G) and y is the zero section. The fact that the evaluation map i s a vector bundle morphism then follows from the fact that 6 and T(<f>) are. (j)**" is a Lie algebra homomorphism : t}>+ i s obviously linear and therefore i t remains to show that i t preserves brackets. Let p e M, suppose (g, p) E D and <f>(g, P) = Q » then i f h e D Pg ^ fl D q we have (h, q) = (h, <f>(g, p)), (hg, p) and (g, p) e D which implies <j>(h, g) = <K'h> c}>(g, p)) = tj>(hg, p) by Def. (4.1) (2). This means $q = c})P<>R(g) on the open set D^" 1 H D q containing e which implies T(cj>q) = T(<j>P) °T(R(g) ) on T(D Pg - 1 Tl Dq) and that T £ ( G ) C T(D Pg - 1 H D q). Then for v e L (G) we have 4>+(v)(<j>P(g)) = <j>+(v)(q) = T(<J>)(v(e), Oq) = T(<j>q)(v(e)) = T(<j, P)oT(R(g))(v(e)) = T(<j>P) (v(g)) which implies that v and 4>+(v) are <{>P-related vector fields. Then [v, v'] i s <f>P-related to [<)>+(v), <j)+(v')] (Bourbaki [ , 8.5.6 p.17]) . Then <{>"*" i s a Lie algebra homomorphism for <f>+([v, v*])(p) = * + ( [ v , v'])(<J>P(e)) = T(<j>P)([v, v'](e)) = [<J>+(v), <J>+(v')]0f>P(e)) = [$ + (vT, <j»-+(v')T(p)' where p was an arbitrary point of M. This completes the proof. ty+ i s called the infinitesimal generator of <j> . If an infin i t e -simal action 8 of G on M i s equal to <f>+ for some local action ty then 0 is called generating. Example (5.4) Let ty be the local action considered in Example (4.4). Let X e L(G) = Hom(E, F). Then cp+(X)(p) = T(«J,)(X(0), 0 p) d_ dt d_ dt <J)(tX, p ) t=0 p o ( t X ° p + I ^ ) - 1 = - p ° X ° p t=0 • - 17 -§6 The Infinitesimal Graph Let 6 : L(G) —> V(M) be an infinitesimal l e f t action and l e t :Pg : G x M —> G and p^ : G x M —> M be the canonical projections. Define f from p *T(G) into ' p *T(M) by f ( g j m ) ( X ( g ) ) = 6(X)(m) where X(g) is the value of X e L(G) at g . (See §2 for definitions of p *T(G) and p *T(M) ). We have f(X(g), 0 J = ( 0 . 9(X)(p)) and f i s \> W P 6 a vector bundle morphism since the evaluation map (X, p) —> 6(X)(p) was assumed to be a vector bundle morphism of L(G) x M into T(M). Then prop.(2.1) implies that the graphs of the .f ^  ^ , t(X(g), e(X)Cp)") : p e M }., are the fibers of a subbundle F f of T(G) x T(M). F f is called the infinitesimal graph of 9 . Proposition (6.1) F f is an integrable subbundle of T(G) x T(M) . Proof : Consider the family of sections ^x^XeL(G) °^ ^ where ? x ( g . P) = (X(g), e(X)(p)). Then (1) by definition of F^ the set {C^(g> P^xeL(G) "*"S tota-'-i n the fiber F^ above (g, p) in F^, and (g,p) (2) i f (X, Y) is any pair of elements of L(G) and i f (g> P) e G x M then [ ? x , C Yl ( g . P) = ([X, Y](g), [9(X), 6(Y)](p)) = ([X, Y](g), 6([X, Y](p» - 18 -since 0 i s a Lie algebra homomorphism. This shows £y](g, p) e and the Theorem of Frobenius (1.5) implies F^ i s integrable. By the definition of integrability there is a fo l i a t i o n Y of G x M such that T(GxM, Y) = F £ . Proposition (6.2) For : g e G, let R(g) be the morphism of G x M into  i t s e l f given by R(g)(h, p) = (hg, p), then R(g) i s also a morphism of Y into Y where Y is the foliation defined by any infinitesimal action 9 of G on M . Proof : T(R(g))(T ( h j p )(GxM, Y)) = T(R(g))({(X(h), 0(X)(p) : X e L(G)» = {(X(hg), 9(X)(p) : X e L(G)} = T(hg,p) ( G X M> Y ) and prop.(1.4) implies that R(g) is a morphism of Y into Y . Remark : Since R(g) i s a diffeomorphism i t takes a maximal connected leaf of Y diffeomorphically onto another maximal connected leaf of Y . The next proposition explains the name "infinitesimal graph". Proposition (6.3) If $ i s any local l e f t action with domain D and  infinitesimal generator (f>+ then the morphism (j>P : g -—> <}>(g, P) of D P - 19..-into M i s an integral (Def. (2 .2 ) ) of f (where f is defined as above  with 8 = cj> ) . Also the graph of is a leaf containing (e, p) of the foli a t i o n Y and the morphism : Y —>. G given by ^pCg* P) = g i s a local diffeomorphism at each point of Y . Proof -: -Let X e L(G) . -Then . T (4>P)(X(g)) = T (4>P)°T (R(g))(X(e)) = Te(<j>-PoR(g))(X(e).) = T e ( < ^ ( g , p ) ) ( X ( e ) ) = *+(X)(Mg, p)) . Hence T <|>P = f. j r ), N. and so d>P is an integral of f . The fact that g (g,<f>p(g)) the graph of <J)P i s a leaf containing (e, p) follows from prop. (2.5). Let (g, p) be any point in Y . Then ir^ is a local d i f f eomor-phism at (e, p) because N^ = {(h, <j)p(h)) : h e D P) i s an open neigh-bourhood of (e, p) i n Y mapped diffeomorphically onto D P by ir^ . Now R(g)(Np) is an open neighbourhood of (g, p) in Y by the remark after prop. (6.2) and -nn is a local diffeomorphism on R(g) (N ) which Lr P completes the proof. We now show that two local actions with the same infinitesimal generator coincide in a neighbourhood of {e} x M . We need a lemma. Lemma (6.4), If an infinitesimal action 8 of G on M i s generating  then,the foliation Y defined by the infinitesimal graph of 0 i s a  Hausdorff manifold. Proof : See Palais [7, Theorem VIII, p.44]. Note : Palais' deM-niL't*ion^of-'leaf--*-di-f-fers"-slightlyfrom'ours; Let <j> and ty be local actions of G on M with domains and .respectively. Let be the connected component of e in D?'- fi D P , then D =• (J? D x. (p) i s an, open neighbourhood of {e} x M * V P'EM P i n G x M (Palais [7, Theorem 1, p.32]). Uniqueness Theorem (6.5) If ty and ty have the same infinitesimal  generator 8 then ty and ty coincide on D . Proof : By prop.(6.3) both ty^ and ty^ are integrals of f (where f is defined as i n prop.(6.3)). Let A C be the set of points on which <J>P and ty^ agree. A i s nonempty since <}>P(e) = ty^(e) = p . Prop. (2.3) implies that A i s open. Let Y as usual be the fol i a t i o n defined by the infinitesimal graph of 8 . A i s closed in since A = $ ^ (A) where $ i s the morphism from into Y x Y given by $(g) = (<|>P(g)> ^ ( g ) ) and A i s the diagonal i n Y x Y which i s a closed set since Y i s Hausdorff (Lemma (6.4)). Then A = D since D i s connected. P P §7' Existence Theorem We now give necessary and sufficient conditions on M for an infinitesimal action of G on M to be generating. Theorem (7.1) A necessary and' sufflcient''condition'that an, infinitesimal  action 8 of G on M is generating i s that the foliation defined by the  infinitesimal graph of 9 is a Hausdorff manifold. Proof : This theorem i s proven in Palais [7, pp.52-58] for f i n i t e dimensional M . The same proof works i n i n f i n i t e dimensions. A weaker theorem giving sufficient (but not necessary) conditions for 9 to be generating i s proven i n Bourbaki" [-2V do-r.'S','- p-.4«84-]*.~ Example (7.2) : Local action generated by an infinitesimal action Consider the infinitesimal action defined i n Example (5.2). Keeping the same notation, let exp : L(G) —> G be the exponential map, then exp = id. If X e V(M), l e t 6 denote the local one-parameter X,t group defined by X . Now i f $> i s a local action of G on M such that (J)* = 9 then <f>(tY, p) = <Kexp tY, p) = 6 + f . (p) by the uniqueness T \*-) jt theorem for d i f f e r e n t i a l equation and definition of <f>+ . Therefore in order to find the local action <j> corresponding to 9 we must find the local one-parameter group corresponding to 0(Y). To shorten notation we w i l l denote (p, p) by p and (p, p)(p, p) by p for p e H . Now we have - 22 -V ) , t ( p ) = 1 - 2t(p, Y) + t Z(p, p)(Y, Y) for ( 1 ) 6e(Y ) ,o ( p > * P ( 2 ) d F ^ . t ^ = 0 ( Y ) ( V Y ) , t ( p ) > Proof of (2) : We have and ^ V ) ' t ( p > = { r - 2 t ( p , Y ) + t 2 p 2 Y 2 } + {2(p, Y) - 2tp 2Y 2} . • 2 V , + o o o o IP - tp Y} {1 - 2t(p\ Y) + t p T J 6 ( Y ) ( 6 e ( Y ) > t ( p ) ) o- 2 { 6 0 ( Y ) f t ( p ) } ( 6 6 ( Y ) f t ( p ) f Y) ~ ( V ) , t ( p ) ' 6 e ( Y ) , t ( p ) ) Y = 2 / P-tp 2Y V l 1 - 2t(p, Y) + t 2p 2Y 2 / P - tp"Y 1 - 2t(p, Y) + t 2p 2Y 2 , Y {1 - 2t(p, Y) + t 2p 2Y 2} 2 (p - tp2Y, p - tp2Y)Y = 2 { p - tp M l - 2t(p, Y) + t 2p 2Y 2} {1 - 2t(p, Y) + t 2p 2Y 2} - 23 -Comparing these two equations we see that (2) is true.. Let D = |(Y, p) e G x M | 1 - 2(p, Y) + (p, p)(Y, Y) t 0 D i s open and contains {0} x M . Finally define <j> : D —> M by A fy T>) = P - (P>- P> Y -n Y ' P ; 1 - 2(p, Y) + (p, p)(Y, Y) * We complete this chapter with a discussion of a special type of ^infinitesimal .action. §8 Uniform Infinitesimal Actions Let 0 : L(G) —-> ~V(M) be an infinitesimal l e f t group action and be the maximal connected leaf through (e, p) of the f o l i a t i o n Y defined by 0 . TT^ : Y —> M is the morphism given by ^gCgj p) = 8 • Definition (8.1) 0 is called a uniform infinitesimal (left) action of G on M i f there exists a connected neighbourhood V of e i n G such that for each p e M the connected component containing (e, p) i n E fl rr ^(y) i s mapped one-to-one onto V by ir-, . V i s called a uniform neighbourhood for 0' . Theorem (8.2) Each maximal connected leaf E of Y is a covering space  for G with covering map TT . = TT^| i f and only i f 0 i s uniform. Proof : Suppose- 8 is uniform. Let V be a uniform neighbourhood. We hav.e to show that for each g c G there exists an open neighbourhood W 'such that T T ^"(W) i s a disjoint union, of open sets in- E- , each of which i s mapped diffeomorphically onto W by TT . We f i r s t show that TT(E) = G : Let (g, p) e E , then by prop.. (6.2), R(g "S-(E). = E^ since. R(g - 1)(g, p) = (e, p)~ and- TT(E)- =• ffoR-(gY(r ) = R(g)°TT_(E ) . So i f P 13 P TT (E ) = G then TT(E) = G also. This w i l l be proven by showing that for G p every positive integer n ; V n CZ TT (E ), then T r 0(E ) w i l l equal G Cr p V J p since any neighbourhood of e in a connected group generates the group. Since V i s a uniform neighbourhood for 8 this i s true for n = 1 . Assume now that V CT . 7 r_(E ), we w i l l show that V C nn(T. ) also. (j p b p > n-l' Let g be any point of V then by the induction hypothesis there exists q-e M such that (g, q) e E P . By prop.(6.2), R ( g _ 1 ) ( E P ) - E Q . Now V C T G(£q) since V i s uniform and so V CZ TrG°R(g~'L) (E p) = R(g _ 1) 0 T r G(^p) • This means gV C TT„(E ) for each g e V , i.e. V CL TT_(E ) . O p b p Now l e t g be any point of G . Let U be a symmetric connected 2 neighbourhood of e in G such that U C V, then W = Ug i s a neighbourhood of g . We w i l l show that TT ^ (W) i s a disjoint union of open sets in E , each of which i s mapped diffeomorphically onto W by TT . Since Tf(E) = G we have TT (^W) i s nonempty. Let C be any component in E of TT 1(W) = ir ^ "(Ug). If (h, s) is any point of C then h e Ug which means gh 1 e U 1 = U and Ugh * C D D C ' V , This implies that Ugh i s a uniform neighbourhood for 8 since V was. Prop. (6.2) gives R ( h _ 1 ) ( E ) = E G since R(h - 1)(h, s) = (e, s) and RCn" 1)(C) is the component of (e, s) i n ,Eg O iT^CUgh x) . But ;%Q maps .the component of (e, s) i n E A Tr~x(Ugh x) diffeomorphically onto Ugh x since s G _1 Ugh i s a uniform neighbourhood which means IT maps C diffeomorphi-c a l l y onto R(h) (Ugh x) = Ug and therefore the pair (E, TT) i s a covering space for G . Conversely suppose TT I : E, —> G i s a covering map. Let V q be any simply connected open neighbourhood of e . Then the component containing (e, q) i n ir ~*"(V) H E" i s a covering space for V and G q therefore must be mapped diffeomorphically onto V . We w i l l need the following theorem i n Chapter 3 . Theorem (8.3) I f G i s simply connected and M i s a Hausdorff manifold  then a uniform i n f i n i t e s i m a l l e f t action 9 : L(G) —> V(M) generates a  global action of G on M . Proof : By the above theorem each leaf E i s a covering space f o r G and since G i s simply connected ^QIJ- : E — > G i s a diffeomorphism. For p e M denote t h i s d i f feomorphism of E onto G by TT? • As usual denote P G by f, the vector bundle morphism from Pg*T(G) into p^*T(M) induced by 0 , and by Y the f o l i a t i o n of G x M defined by the integrable subbundle F f : Define <J>P : G —> M to be <f>P(g) = * o (Tr P) - 1(g) where TT m : Y —> M i s TTM(g> m) = m. F i n a l l y define tj> : G x M —> M to be (|>(g, p) = <j>P(g). We w i l l show that <j> i s a global group action with i n f i n i t e s i m a l generator - 26..--G . Let v E L ( G ) . Note that each cj>P is an integral for f since T((j>P)(v(g)) = TCTr^oTCTT^^CvCg)) = T(TT M ) (v(g), f n , (V(g))) •(g,o£) (g)) = f n (v(g)) as ( T r P ) _ 1 ( g ) = <j>P(g) . (g,*P(g)> ty is a global action : (1) ty(e, p) = TT m° ( T T P ) 1(e) = Tr M(e, p) = p since (e, p) is the unique point in E^ with f i r s t component equal to e . (2) Show ty(g, ty(h, p)) = ty(gh, p) for a l l g, h e G and p E M. Define ty-^(g) = <Kg> <J>(h, p)) and = "f^S*1, P) • B y t' i e definition of ty , the graph of ty^ is p) a n c* t^ i e g r aPh of ty^ is R(h _ 1)(E p) = l since R(h _ 1)(h, ty(h, p)) = (e, <j>(h, p)). Then since ^ K h j p ) i g o n g _ t o _ o n e o n E ^ ^ ^ we have <f>(g, <f>(h, p)) = tyigh, p). We now show that ty : G x M —> M i s a morphism. For p E M , define P f A = < g E G : there exists some open neighbourhood U of g and some open neighbourhood V of p such that ty i s a morphism on U x V j- . - 27 -, (a) A F contains e and'therefore Ap £ 0 ' . Let p : M —> M be the identity. By prop. (2.6) there exists a connected open neighbourhood U x V of (e, p) in G x M and a morphism if) : U x v —> M such that for a l l m e V the morphism. i p m : g — > i|>(g, m) is-an integral for f with i|>m(e") = p(m)- = m- .» <j>™ is- also an integral for f on U x v with <J>m(e) = i^ m(e). . Since- M i s Hausdorff and U i s connected i t follows, just as in the proof of theorem (6.4) using the uniqueness of integrals, tHat <j>m = ^ m on U ; i.e. § = on U x V and A? . contains e . (b) A P is open i n G by definition. (c) A? i s closed in G .• Let g e A P , by (a) above there exists a connected neighbourhood U x V of (e, <J>(g, P)) such that <}> is a morphism on U x V . We denote <|> by g on U x v to emphasize that i t i s a morphism. Furthermore we assume U = U x . Since h —> (j>(h, p) i s an integral, and so in particular continuous, there exists a neighbourhood N of g such that c|>(N, p) CI V. Let h e N f\ Ug H A? ; h exists since g e A P and N O Dg i s a neighbourhood of g . By the definition of A P there exists a connected neighbourhood x of (h, p) on which <j> i s a morphism and since <Kh, p) e V we can assume (shrinking i f necessary) that <f)(h, V^) C V . Define y : Uh x V x —> M by y(k, m) = 6(kh~ 1, <J>(h, m)). y i s a morphism on Uh x since i t is a composition of morphisms; y = go(R(h x) x . Now for m e V^, we have the morphism ym : U..J, —-> M given by - 28 -y m(k) = y(k, m) with y(h, ro) = -<j)(h, m) . We w i l l show that y m i s an integral for f and then (as in the proof of (a)) since <j>m i s also an integral with the same value at h, <f>m w i l l equal y m and <J> w i l l be a morphism on Uh x . Let g m : U —> M be the morphism defined by B m(k) = B(k, m). Since g = ty on U-x V we: have g m = <f>m on U . Let X e L(G), to show that y is <an integral'we need that T (y m) (X(g)) = f m (X(g).) = . e(x.).(.Ym(g)) . g (g,Y (g)) We have T ( Y m)(X(g)) = T ( B ( | , ( h' m )oR(h" 1))(X(g)) (by def. of y) o o = T ( B < | ) ( h' m ))(X(gh" : i :)) (s'ince X e L(G)) • gh" 1 = 8 ( X ) ( ^ h ' m ) ( g h _ 1 ) ) (since B* ( h' m ) = * K h ' m ) i s an integral of f ) l ( X ) ( B * ( h ' m ) o R ( h - 1 ) ( g ) ) i(X)(y m(g)) and so y m i s an integral of f . Therefore ty is a morphism on Uh x and h e Ug implies g E U \ = Uh , i.e. (g, p) E Uh x and g E A P showing that ' A P = A P . Since G is. connected (a), (b) and (c) imply that A P = G . As p was arbitrary ty is a morphism on G x M . It remains to show that e i s the infinitesimal generator of ty. - 2.9 -Let X e L(G), then Te<j>P(X(e)) = f: (X(e)) (since <j-P is an (e,(j)P(e)) integral for f) 6(X)(<J,P(e)) = 0(X)( P) showing that 9 is- the- infinitesimal- generator-of? <J> and completing the proof of the theorem. The proof that <j> i s a morphism is essentially the same as the proof showing that the flow of a vector f i e l d i s a morphism. (Cf. Lang [5, p.80]). Proposition (8.4) Let ty : G x M —> M be a global l e f t action of G a Hausdorff manifold M. Let X e L(G) and {6t} be the one-parameter  group corresponding to <j>+(X). Then S t(p) = <J)(expGtX, p) for a l l p e M Proof : <J)(expG0-X, p) = A ( e , p) = p and • — <J.(expGtX, p) = jj^ <j)(expG(s+t)X, p) s=0 on d 1 (f>(expGsX, c()(expGtX,. p)) s=0 since .ty is a global action ds + • + = ty (X) (ty (exp GtX, p)) by definition of <j> The result then, follows from the uniqueness theorem- for- d-if"ferential equations. - 30 -Chapter 3 Connected Lie.Transformation Groups Let D(M) be the group of diffeomorphisms of the manifold M .. A Lie group G is- called a- Lie- transformation group of M;> if-the-underlying group of G is a subgroup of D(M) and i f the map (gj P) —> g(p) of G x M into M i s a morphism. Of course one could give G the discrete topology and this would automatically be true. A nontrivial example is the group I(M) of isometries of a f i n i t e dimen-sional Riemannian manifold, which is a Lie transformation group with respect to the compact open topology. Further examples of Lie transforma-tion groups can be found in H. Chu and S. Kobayashi [3]. The main result of this chapter is to show that there i s a one-to-one correspondence between connected Lie transformation groups of M and certain subalgebras of the Lie algebra of vector fields V(M) where M i s a Hausdorff manifold. In this chapter M w i l l always denote a Hausdorff manifold. §9 The Image of the Infinitesimal Generator of a. Lie Transformation Group Let G be a Lie transformation group of M . Then there is a global action of G on M with infinitesimal generator (j) : L(G) —> V(M) . We now examine the image of (j> i n V(M). Let exp : L(G) —> G be the exponential map. - 31 -+ Proposition (9.1) The image <f> (L(G)) consists of complete vector fields and the one-parameter group corresponding to <f> (X) is exp tX . Proof : exp tX .is a one-parameter group and d_ dt exp tX(p) = cf, (X)(p) t=0. The result then follows from the uniqueness theorem for di f f e r e n t i a l equations. + Proposition (9.2) <f> is injective. Proof : If <j,+ (X) = 0 then d_ dt exp tX(p) = t=s dt t=0 exp((s + t)(X)(p) d_ dt t=0 exp tX(exp sX(p)) = <j> (X)(exp sX(p)) ' = 0 exp sX(p) for a l l p e M . This means exp tX(p) = p for a l l t e R and a l l p E M, i.e. exp tX = id which implies that X = 0 since exp has a radius of i n j e c t i v i t y at 0 in L(G). M Proposition (9.3) $ (L(G)) 'possesses a Banach Lie algebra structure such  that the evaluation map (Y,. p) —> Y(p) is a vector bundle morphism from - 32 -the t r i v i a l vector bundle <J>-' (L(G) ) x M into T(M) and (j>+ '• L(G) —> <j>+(L(G)) i s a Banach Lie- algebra isomorphism. Furthermore, -this Banach space structure is necessarily unique. Proof : By prop. ( 9 . 2 ) <j>+ : L(G) —> <j>+(L(G)) is a Lie algebra isomor-phism and hence induces a Bariach Lie algebra structure on <f>"*"(L(G)) making + + <J> a Banach Lie algebra* isomorphism; Define 3 : <j> (L(G)) x M —> L(G) x M H — 1 by 3(Y, p) = ((<f> ) (Y), p) , then 3 is easily seen to be. a vector bundle morphism. Now prop. ( 5 . 3 ) gives.that the map a : (X, p) —> <j>+(X) (p) of L(G) x M —> T(M) is a vector bundle morphism. The evaluation map + (j> (L(G)) x M —> T(M) is equal to a°3 and therefore is a vector bundle morphism. The uniqueness of the Banach space structure comes from the following proposition. Proposition ( 9 . 4 ) Let E be a vector bundle over M and l e t V be a  vector space of sections :of E . If V admits two Banach space structures  such that the evaluation map (X, p) — > X(p) of V x M. into E i s continuous with respect to both then the identity map from V into V i s a homeomorphism, i.e. the two norms are equivalent. Proof : Let and denote V with respect to the two topologies and let e i : x M —> E ( i = 1, 2) be the evaluation maps. By the closed graph theorem, i n order to show that id : ••—-> V 2 is continuous, i t i s enough to show that the diagonal i s closed in x . Let 1 2 { ( X ^ , X r) } be a Cauchy sequence in the. diagonal of x V^, i.e. 1 2 1 2 X„ £ V,, X E V„ and X = X . Since V, x V. is complete there exists n l n z n n . 1 2 - 33 -a limit point (X, Y) of this sequence; but for a l l p e M we have . X(p) = lim e^X*, p) = lim X*(p) n-*» n-*» = lim X 2 C p ) n n-x» = ,lim-e2.(X2, p) = Y(p) . n-*=° Therefore X = Y and the diagonal is closed in x . Interchanging and V 2 above gives that id : —> i s continuous also and id is a homeomorphism. If Y is a complete vector f i e l d then denote by Exp tY, the one-parameter group generated.,by Y. Let <|>. : G x M —> M be the global action of a Lie transformation group G. Prop.(9.1) gives that exp_(X) = Exp(<j>+(X)) and this implies the following result. Proposition (9.5) Exp i s injective on a neighbourhood of 0 i n A +(L(G)) in the topology induced on <|)+(L(G)) by <j>+ . §10 Banach Lie Algebras of Complete Vector Fields We now consider when a Lie subalgebra L of V(M) is the image of the infinitesimal generator of a connected Lie transformation group. In view of propositions (9.1), (9.3), and (9.5) we only consider Lie subalgebras L of V(M) which satisfy the following conditions . (A) L consists of complete vector f i e l d s ; - 34 -(B) L has a Banach Lie algebra structure, (necessarily unique by Prop. (9.4)) such that ; (Bl) the evaluation map ev : (X, p) —-> X(p) is a vector bundle morphism from the t r i v i a l bundle L x M into T(M). (B2) there exists an open ball- B^ (O-) of. radius r at 0 such that Exp : L —> D(M) is infective on B r(0). •Proposition (10.1) If L is f i n i t e dimensional and satisfies (A) then condition (B) is true also. Proof : Since L is f i n i t e dimensiional i t has a natural Banach space structure. A proof of (Bl) is in Bourbaki [2, Remarque p.140] and a proof of' (B2) is in Loos '[6, p. 182]. The rest of this section w i l l be devoted to proving the following theorem. Theorem (10.2) If M is a Hausdorff manifold and L is a Lie subalgebra  of V(M) satisfying conditions (A) and (B) then there exists a unique  connected Lie transformation group G with natural global action (J) : G x M —> M such that <j>+ is a Banach Lie algebra isomorphism of L(G) onto L. Remark : Palais [7] f i r s t proved this theorem in the case where L and M are f i n i t e dimensional. Using a different method, Loos [6] extended this result to the case where L i s f i n i t e dimensional and M i s a (not - 35 -necessarily Hausforff) Banach mainfold. The proof given here i s similar to Palais' . In order to prove Theorem (10.2) we need the following theorem, which i s of interest in i t s e l f . Theorem (10.3). If II is a Hausdorff manifold and L is a Lie subalgebra of V(M) satisfying condition (A) and admitting a Banach Lie algebra structure such'that (Bl) js true (but not necessarily (-B2)) then there exists  a simply connected Lie group G with L(G) = AL and a global action ty : G x M —> M such that ijj +(L(G)) = L, ty + is a continuous linear map  into L, and for C e AL ; ty+(C) = C(l). Proof : By theorem (3.1)-there exists a Lie group with Lie algebra AL. Let G be the universal covering group (see Bourbaki [2, p.113]) of this group ; then L(G) = AL. We have an infinitesimal l e f t action, which we c a l l ty+, of G on M given by the following sequence of vector bundle morphisms, ev AL x M -—> L x M > T(M) (C, p) > (C(l), p) > C(l)(p) where ev i s the evaluation map which is a vector bundle morphism by condition (Bl). The map ty+ : C —•> C(l) i s continuous by theorem (3.1). The existence of the global action ty w i l l follow from theorem (8.3) by showing that ty+ is a uniform infinitesimal l e f t action. - 36 -By condition (BI), ev : L x M —> T(M) is a vector bundle morphism. Then the global version of the existence theorem for d i f f e r e n t i a l equations depending on a parameter (in this case the parameter space is L) implies that the map (t, X, p) —> Exp(tX)(p) from (R x L x M into M i s a morphism. The fact that this flow is defined on a l l of IR follows from condition (A) . Now let B (0) be an open b a l l about 0 in AL on which, exp-p G is a dif feomorphism. We w i l l show that i|> i s uniform on V = exp^(B (0)) G p P + -1 Define, for each p e M, the map 6^ : g —> (g, ExpOJi (exp ? (g)))(p)) G from V into G x M . Let £ p be the maximal connected leaf containing (e, p) i n the foliation Y of G x M defined by the ••infinitesimal graph of 4 . For X e L(G) -and p e M define aX ! t ~~~> ( e xPG^ t X)» Exp(t^ +(X)(p)) from IR into G x M . Now T(a P) d Ut s da X dt dt t=0 ex Pg(tX)expg(sX), ~ Exp (tip (X))(p) t=s X(expg(sX) , i|»+(X) (Exp(si|; +(X)) (p)) belongs to T(G x M, Y). Then prop.(1.4) implies that the image of a P i s in Z p since a P(0) = (e, p) and IR i s connected. In particular i f - 37 -g•= expx(X) for X e. B (0) then-^ P •a P(l) = (expg(X), Exp(./(X))(p)) = (g, Exp(4 ) +(exp~ 1(g))(p)) which shows 'that S P ( V ) CT ^p * Denote by V P the image of V under 5 P . The "projection" T T g : Y — > G obviously maps V P onerto-one onto V . In fact we now show that V P is the component containing (e, p) in TT^(V) fl E . We have b P (e, p) = 6 P(e) e V P . Suppose (g, q) i s any point in V P and l e t U be ah open set in E containing (g, q) on which TT~ i s a diffeomoephism P G (prop. (6.3)). Let W be'ah- open-set in' V ft ir~(U')' cO'nta'ihl'ri'g g" , then TT~^ : Tr~(U) —> U takes W onto an open set containing (g, q) and -1 G^ irft(U) (W) C V P since TT-1 = 6 P on W . This proves that V P i s open. In order to show that V p is the component containing (e, p) in n>,^ (V) H E i t remains to prove that V P is closed in ir=^(V) . Let Or P \J (h, m) be any point in irx^(V) D E such that (h, m) i V P . Now since G p IT- i s one-to-one on V P there exists a unique point in V P with f i r s t component h, say (h, n). E^ is Hausdorff since i t i s an open submani-fold of Y and therefore we can find disjoint open neighbourhoods A and B of (h, m) and (h, n) respectively which TT~ maps diffeomorphically onto the same neighbourhood of h ; this i s possible since Tr= is a local diffeomorphism. By restricting A and. B further we can assume that • A C. V P and i t then follows that B fl V P = 0 since TTX is one-to-one on - 38 -V1' . This completes the proof that if/ i s uniform and proves the theorem. Proof of Theorem (10.2) We keep the notations used above. L i s assumed to be a subalgebra of V(M) s a t i s f y i n g conditions (A) and -(B); Consider the i d e a l ker if) = { C e L(G) : C ( l ) =0} . i n L(G) which i s the kernel of the map ip + : L(G) — > V(M); i t i s closed i n L(G) since ker if; + = C\ (if;+) x ( 0 ) where, ifj~|~ i s the continuous l i n e a r map peM p P P X — > ij; +(X)(p) from L(G) i n t o T p(M) ( 0 p denotes the zero vector i n Tp(M) ). Therefore ker ifj + i s a Banach L i e subalgebra of L(G) . Let L' = j C £ L(G) = AL : C(t) = tX for some -X e L then L' i s a closed vector subspace i n L(G) which complements ker if; , i . e . ker if)+ s p l i t s i n L(G) and we i d e n t i f y L ( 5 ) with ker ifj + x L 1 . For g £ G, denote by if; the d i f feomorphism p —> iKg, P) of M in t o M. Let 6 : G — > D(M) be the group homomorphism g —> ifj . Let H = ker 6 , then we have a group isomorphism 6 : G/H — > 6(G). We w i l l show that H i s a L i e subgroup of G. Condition (B2) gives the existence of a open neighbourhood N of 0 i n L on which Exp i s i n f e c t i v e . Let A x B C ker if;+ x L' = L(G) be an open neighbourhood of 0 on which exp~ i s a d i f feomorphism and such that if)+(A x B) C N. Let h E H f\ exp- ( A x B) , G then h = expg (C) for some unique C E A x B. Now f o r a l l p E M we have - 39 -P = K » , P) = ^(exp~ CC), p) = Exp OJi+(C))(p) Cby prop. (8,4)) = Exp (C(l))(p) . Since C(l) e N and Exp (C(l)) = id we have C(l) = 0, i.e. C e A x {0} . Also i f C e A x {0} then iJj(exp(C), p) = p for a l l p and exp--(C) E H. The fact that H i s a Lie subgroup then follows from prop. (3.3) since expg (A x {0}) = H H expg (A x B). We also have L(H) = ker if)+ . It follows from prop. (3.6) that there exists a connected Lie group structure on G/H such that the projectiion p : G —> G/H i s a submersion and L(G/H) = L(G)/L(H) = L . Using the group isomorphism 6 : G/H —> 6(G) we have a Lie group structure induced on 6(G) such that 6 is a submersion, ker L(6) = ker \p+, L(6(G)) = L, and 6 is a diffeomorphism. With this Lie group structure 6(G) w i l l be denoted by G. Define <J> : G x M —> M to be the natural action <j>(g, p) = g(p). Let a be the submersion (k, p) —> (6(k), p) of G x M into G x M. Then if) = <j>oa and <j> i s a morphism since if) is a morphism and a i s a submersion. We now show that cf>+ i s a Banach Lie algebra isomorphism. Let C e L(G) = AL and p e M, then C(l) (P) = if)+(C) (p) = T ( e j p ) C ) ( C ( e ) , 0 p) - 40 -T ( e > p ) ( * o a ) ( C ( e ) , 0 p) T ( i d > P ) ^ T e 5 C C ( e ) ) ' V <J>+(L(S)(C))(p) , i.e. ty+ = <j>+°L(5). • + + We see that ty maps L(G) onto L- since ty- maps L(G) onto L + + (theorem (10.3)). Since L(6)(kar ip< ) = 0 we have that .-ty is infective. The fact that ty+ i s continuous' follows easily from the following; (1) ty+ i s continuous + (2) L(5) is continuous, surjective and ker L(6) = ker ty s p l i t s in L(G) (since 5 i s a submersion). It remains to prove the uniqueness of G. Let F be another Lie transformation group with the same properties as G and let f3 : F x M —> M be the map (f, p) —> f(p). Now exp„ (tX) = Exp (tB +(X)) for X e L(F) by prop. (9.1) and therefore since F i s connected i t i s generated by Exp (L). Similarly G i s generated by Exp (L) which shows that the underlying groups of G and F are the same in D(M). The following commutative diagram ( e W L(G) — > L(F) exp G e x P p - 41 -shows that Id from G into F is a morphism and completes the proof of the theorem. §11 A Banach Lie algebra of complete vector fields which does not generate a connected Lie transformation group If we have a subalgebra L of V'(M). which satisfies condition (A) and admits a Banach Lie algebra structure such that (Bl) i s true but not (B2) then prop.(9.4) implies that this L won't admit any Banach space structure such that (B2) i s satisfied-. Hence by prop. (9.5), L isn't the-image of the infinitesimal generator of any connected Lie transformation group. We now give an example of such an L which, although i t doesn't generate a connected Lie transformation group, i s s t i l l enlargeable. Let M = disjoint KJ S where is the unit c i r c l e S . Define the vector J • , n n=l f i e l d X by n xn(P) -0 i f p e s\ and i ^ n P 3 , t, „l 2irit where a is the curve on S ; t —> e n Let L be the normed vector space consisting of a l l sums co co | c | • T c X , c e IR, such that Y — — <<*>. If C = J c X E L , define tl-, n n n L. n L n n * n-1 n=l - 42 -0 3 c i i i i r n the norm of C to be | | C | ] = I . We w i l l show that L i s a n=l Banach Lie algebra satisfying (A) and (Bl) but not (B2). L consists of complete vector fields since we have a disjoint union of compact manifolds. It is a vector space for lc + b I MI c X + I b X || = I n n ' 1u n n L n n' 1 L n |c | |b | < oo n u n i f 7 c X^ and I b X belong to L. Similarly L i s closed under u n n u n n ° J scalar multiplication. L i s closed under the bracket operation since ll c X , J b X- ] = V c.b.[X., X.] = 0 u n n L n n .L. I i i ' i We now give the usual proof that a space of sequences i s complete. Let {An} be a Cauchy sequence of elements in L, i.e. given e > 0 there exists N such that i f i , j > N we have M A 1 - A^ll =1 < e k k where A n = £ ^ y^k ' ^n P a r t i c u l a r this implies that for fixed k, k is a Cauchy sequence. Let A^ = lim A^ and A = ^ A^X^ . We w i l l show that A e L and lim A n = A . From above we have n-x» - 43 , -for a l l s > 1 and i , j > N , then lira 2, = 2, Z—— < °°- f° r a*-*- s > 1. k=l k 1=1 k Since this i s true for a l l s we have k=i k But this implies that A - A"' e L which means A = (A - A~*) + A^ e L . We also have for j > N , | |A - A~* | | < E ; which shows that lim A~* .= A We have now shown that L i s a Banach Lie algebra and i t i s t r i v i a l l y enlargeable since i t is abelian. It remains to show that the evaluation map ev : L x M —> T(M) is a vector bundle morphism. Local coordinates on each of the 's are n given by the local inverse of the map t —> e^ 1*" . We denote this map by l ° 8 n • This induces local coordinates on T(M) and we denote this map T by i ° 8 n • If TT • T(M) —> M i s the usual projection and i f Z = £ z n x n then log^(Z(p)) = (log n (Tr(Z(p)), z n) = (log n(p), z n) . Denote by Trn ' the continuous linear map from L into R given by £ a]Xy_ — > a n • A local coordinates map at (Y, p) z L x M is given by (Y, p) —> (Y, log n(p)) i f , p e SJ . Let Y = £ and W be an.open neighbourhood of Y(p) = ev(Y, p) in T CM). Now log^(Y(p)) = (log n(p), y ) - 44 -and by the definition of the topology of T(M) there exists an open set U containing log (p) and an interval B (y ) about y such that ° . n p n n ( l o g 1 ) " 1 ^ x B (y )) C W . Let Z = Y z X , i f v bn p •'n ^ . L n n ( Z , q) e B p / n(Y)' x (iog n) _ 1(U) C L * M t h e n l l z ~ Y ' l I < p / n which implies, \'z^ - y^J v , <~ p,*. whic,.^ . lrapLi,es,^ lp,g^ .(-§.CaX).» x~ B^ .Cy.^ ),;, which shows ev(B , (Y) x TJ) C~ W . Hence ev. i s continuous and these local coordinates v p/n — ev is given By the map r in the following diagram, (Y, p) 2 L _ > Y(p) (Y, log n(p)) — — > (log n.(p), TTn(p).) and x is a morphism since TT i s . This proves that ev i s a morphism of manifolds. It i s a vector bundle morphism since the constant map p — > TT^ i s a morphism from U into the space of continuous linear maps from L into (R . Condition (B2) doesn't hold because Exp doesn't have a radius of i n j e c t i v i t y at 0 in L, for Exp X^ = identity for a l l n and lim X = 0 . n n-*» Although L doesn't generate a connected Lie transformation group theorem (10.3) ensures that.it i s the image of the infinitesimal generator of a global l e f t action. - 45 -Bibliography N. Bourbaki, Varietes differentialles et'analytiques, Hermann, Paris, 1971. N. Bourbaki, Grpupes et, algebres ,de Lie, Hermann, Paris, 1972, chapters 2 and 3 . H. Chu and S. Kobayashi, The automorphism group of a geometric structure, Trans. Amer.. Math. Soc. 113 (1964) pp. 141-150. H.T. van Est and Th. J. Korthagen, Non-enlargeable Lie algebras, Indag. Math. 26 (1964) pp. 15-31. S. Lang, Dif f er en t iab.l e Mani f olds, Addison-Wesley^ Reading-, Mass., 1972. 0. Loos, Lie transformation groups of Banach manifolds, Journal of Differential Geometry 5 (1971) pp. 175-185. R.S. Palais, A global formulation of the Lie theory of transformation  groups, Memoirs of the Amer. Math. Soc. No. 22, 1957. S. Swierczkowski, The path-functor on Banach Lie algebras, Proc. Kon. Ned. Akad. v. Weterrsch., Amsterdam, 74 (1971) pp. 235-239. 

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