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Periodica M atl~ematica Hungarica Vol. 13 (4), (1982), pp. 289--308
TRANSFORMATIONS OF LINEAR CONNECTIONS
by
E. VASSILIOU (Athens)
Abstract
Combining several results on related (or conjugate) connections, defined on banachable fibre bundles, we set up a machinery, which pemnits to study various trans- formations of linear connections. Global and local methods are applied throughout. As an application, we get an extension of the classical affino transformations to the context of infinite-dimensional vector bundles. Another application shows that, l'ealising the ordinary linear differential equations (in Banach spaces) as connections, we get the usual transfol~nations of (equivalent) equations. Thus, some classical results on differential equations, such as the Theorem of Floquot, can have a "geom¢~tric" interpretation.
§ O. Introduction
The principal object of this no te is the s tudy of related l inear connections oll banachable f ibre bundles (the te rms conjugate or adap ted connections are also used, cf. e,g. [7], [11], [12]). Rough ly speaking, two connections are re la ted by an appropr ia te bundle morphism if the la t ter preserves the corre- sponding horizontal subspaces.
I t is well known t h a t l inear connect ions can be s tudied e i ther d i rect ly on vec tor bundles, or as infini tesimal connections on the corresponding f rame bundles. Here, combining several results on re la ted connections and using both the above formalisms, we obta in a general machinery , which permi ts to der ive var ious t rans format ions o f connections.
The local and global tools of the t heo ry of connections are used through- out . 'Global methods are main ly based upon split t ings of exac t sequences, connect ion maps and connect ion forms. L o c a l methods , on the o ther hand, are based on the (infinite-dimensional) Christoffel symbols and the local connect ion forms. I t is ev iden t t h a t such a va r i e ty of means provides a v e r y flexible machinery , which can be applied according to the p roper problem and the f r amework under considerat ion. The main results o n t ransformat ions are Theorems 4.3 and 4.6.
In the sequel, in order to i l lustrate the general ideas of this note, we give two applicat ions: the f irs t shows how the classical not ion of aff ine t rans- fo rmat ion (defined for connect ions on t a n g e n t b u n d l e s ) c a n be ex tended to the general con tex t of a r b i t r a r y banaehable vec to r bundles. W e obta in t he i r character is t ic p roper ty , s ta ted in Theorem 5.5. T h e s e c o n d appl icat ion is on
A M S (MOS) subject classi/icatlons (1980). Primary 58B20; Secondary 53C05~ Key words and phrases. Linear connections, connections on banachahle principal
bundles, Christoffel symbols, connection forms, transformations.
290 v A s s ~ o u : Ti~ANSFORMATION8 OF LINEAR OONNEU~ION8
trivial bundles. More precisely, we show that , if we interpret the ordinary linear differential equations (in Banach spaces) as connections on trivial bundles, the usual transformations of (equivalent) equations correspond, naturally, to related connections. This gives rise to a "geometric", i.e. using the language of connections, formulation of some classical results on equations, such as the Theorem of Floquet (cf. Corollary 5.8). Although we do not go into further details in this direction, we note that this point of view may yield a classifica- tion ef the connections analogous to that for equivalent equations.
As we have already mentioned, our context is the theory of connections on banachable fibre bundles. However, in some instances (e.g. Corollary 2.4), we indicate how the general formalism of this note yields the classical results and formulas, found in standard references.
§ 1. Notations
In this note manifolds and fibre bundles are modelled en Banach spaces and, for the sake of simplicity, differentiability is of class C*% Although we use the standard terminology of [1], [2], [8], [9], we fix the following notations, which appear very frequently throughout.
We denote by l ---- (P, G, B, ~t) an arbitrary principal [/bre bundle (abbre- viation: p.b) and by L = (E, B, p) a vector bundle (v.b). For the local structure of vector bundles we follow the formalism of ([1]; p. 12). That is, if (U, ~) is a local chart of B, the corresponding v.b-chart is a triple (@,~, U) with
:p - l (U)_% ~(U)X F. Thus, if s: U -~ E is a local section of L, the principal part s~: ~(U) --~ F, with respect to the previous charts, is given by
(1.1) ( ~ O 8 O ~0--1)(X) ~--- (X, 8~(X)); X E ~o(V).
Similarly, for a vector field X of B, X~ : ~(U) -* B is given by
(1.2) (Tv o X o = ( . , X . ( . ) ) ; • E v(U).
Assume now that (F, h) is a v.b-morphism of L into L ' = (E', B', p'). For suitable charts (O, ~, U) and (@', ~', U'), there exists a smooth morphism
(~ ' o F o @-1)~ : ~(U) ~ L(F, F')
such that
(1.3) ( ~ ' o F o ~ - l ) ( x , u) = ((~' o h o ¢-~)(x), ( ~ ' o F o ~ - s ) u ( x ) u ) ,
for every x E ~(U) and u E Y. For simplicity of notations, we set
(1.4) h~ : : ~' o h o ~ -1
(1.5) (g,p) ~ : : (4[:" o F o ~ - ~ ) ~ .
VASSILIOU: TRANSFORMATIONS OF LINEAR CONNEC~IONS 291
Let now L be a vector bundle of standard fibre type F and let us denote by ~(E) the subbundle of the linear map bundle ~(BX F, E), with total space P(E) consisting of all the pairs ra:----(b,r ) E {b}XLis(F, Eb), where b is running in B and E b : = I~-1(b) • The triple ~(E) = (P(E), B, q), with q(r~) = b, is a vector bundle (in fact an open subbundle of £ (BXF, E)). For the corre- sponding chart (U, 9) of B and (~, 9, U) of L, we have the v.b-chart of ~(E)
given by
(1.6)
L, : = ~-~(U) -~ 9(U)×Lis (F)
L~(rb) : : (9(b), ~b o r),
where r b Eq-I(U) and ~b : E b - ~ F " We get indeed a v.b-structnro on £(E), as we can easily check, by a proper modification of the general construction of ([1]; p. 2 1 )
Moreover, the quadruple l(E) = (P(E), GL(F), B, ~), where ~(rb) : b, is a principal bundle, the bundle of frames o i l ([2], p. 85). The action is given by % ' 9 : - - - - ( b , r o g), for every r b EP(E) and g E GL(F). For local charts as above, we have the local trivialisation
given by
(1.7)
l, : ~-I(U) --. U×GL(F),
l~(rb) : : (b, ~b o r); rb E ~ - I ( U ) •
In particular, the inverse of l¢ satisfies the identity
(1.8) l$X(b, g o g') = l$ l (b , g) • if',
for every b E U and g, g' E GL(F). As in the case of vector bundles, if R : U -~ P(E) is a local section of
I(E), we define the principal part R~ : 9(U) ~ Lis(F), with
(1.9) R~ : = F ' , o l~ o R o 9 -1.
Finally, we recall that L is associated with l(E) via the frame map : P(E) X F -~ E, given by
(1.10) 0(rb, U) : = r(u), .....
if r b = (b, r) and u E F. I t is also known t h a t
(1.11) u) • a) : = r o a), g ' l (u ) ) = r(u),
for any g E GL(F).
292 ~ASSILIOU:'TRANSFORMATIONS OF LINEAR CONNECTIONS
§ 2. Connections on fibre bundles
Le t l ~ (P, G, B , ~z) be a principal bundle. A (smooth infinitesimal) connection on l is a G-split t ing of the exact sequence of vec to r bundles
(*)
where:
(;
0 -* P × (; .L~ T P ~-~ ~z* ( T B ) -~ O,
is the Lie Banach algebra of the s t ruc tura l group G; is the v .b-morphism def ined by ~(p, X) : - - T~p,e)~(X), for eve ry
P E P and X E ( ; ~ T e G ; is the (right) act ion o f G on P ;
: z* (TB) is the pull-back of T B via u: T:~! is the v .b-morphism def ined by the universal p rope r ty of pull-
backs.
Thus, there exists e i ther a G-v.b-morphism c : ~ * ( T B ) ~ T P so t ha t Txg ! o c ~- id~.(TB) , or a G-v:b-morphism V : T P ~ P × (; such t ha t V o v -- -:- idp ×G. The spli t t ing maps c and V are G-morphisms in the sense t ha t t hey preserve the act ion of G on the bundles of (*), def ined in a na tura l manner .
As a consequence. T P ~- V P s H P where V P : : Ira(u) and H P :- - ~- Im(c). Hence, each vec to r u E T P has the unique decomposi t ion u -- u ~ + u h, t h e exponents v and h denot ing respect ive ly the vertical and horizontal com- ponents .
The spli t t ing morphism V defines, equivalent ly , the so-called connection form to, i.e.. a ( ;-valued differential 1-form on P satisfying the well-known proper t ies (cf. e.g. [8; p. 64]). Here we have t h a t
I t is easy to check t h a t
(2.1)
% ( u ) - - (pr , o V) (u); u E T~P.
for eve ry p E P and u E T p P . Let us consider now an open covering ~ : {Ut}ie I of B and a f ixed
fami ly of local sections {si}i~ I of P , over C. As in the f ini te-dimensional case, co gives rise to the local connection forms ~o i :-~ sio~ (i E I ) . The converse is also t rue . i f a cer ta in compat ib i l i ty condition, on over lapping sets o f ~, holds. :For details we refer to ([8], pp. 65 66; [15], pp. 226 228), whose construct ions car ry over to the infini te-dimensional context .
Le t now L -- (E, B, p) be a vec to r bundle. A (smooth) connection on L is a spli t t ing of the exac t sequence of vec tor bundles
(**) 0 ~ V E i-L'~ TETP-~t p * ( T B ) --* O,
VASSILIOU: TRANSFORMATIONS OF L I N E A R ' CONNECTIONS 2 9 3
where VE is the vert ical subbundle of TE, j the canonical v.b-morphism and the other symbols have a meaning analogous to t ha t of their counterparts in (*). We denote by 7 : p*(TB) -* T E the spli t t ing morphism with Tp ! o 7 - - idp.(TB), and by W : T E -* VE the spli t t ing morphism with W o ~ -- idvE.
We have the decomposit ion T E -- V E ~ H E , where H E :-- Im(~). Equivalent ly , a connection on L is def ined by the connection map
K : T E -~ E (ef. [20]). In particular, if K is linear on the p.-f ibres , the con- nection is said to be linear. In this case we define the local Christoffel symbols
r~ : ~(U) ~ L2(Fx B, F),
which are smooth morphisms satisfying the well-known compatibi l i ty condi- tions (cf. [6]; p. 5). Sett ing K~ :-- • o K o T O -1 (the local expression of K) we have t h a t
(2.2) K~(x, u, y, v) = (x, r~(x)(u, y) + c),
for every (x, u, y, v) E ~ ( U ) × F X B x F . I t is also known tha t a linear connection 7 on a vector bundle of s tandard
fibre F derives from an (infinitesimal linear) connection c( ~- co) on the bundle of frames l(E). The corresponding split t ing mori)hisms 7 and c are related by (of. also [13]):
(2.3) 7(e, v) = T~,~)~.(c(r~, v)),
for every (e, v) E E × , T B with e -- ~(r b, u) -- r(u). Formula (2.3) enables one to connect two basic local ingredients of the
linear connection theory; namely, the Christoffel symbols and the local con- nection forms. I f s : U ~ E is a local section of L, we can determine a smooth morphism a : P ( E ) * F such tha t s (n (p) ) -~ ~(p, a(p)), f i r every p E P(E) (cf. [2]; 6.5.6). Thus, i f R : U --~ P(E) is a local section of l(E), we have tha t
(2.4) s(b) -~ q(R(b), a(R(b))); b E U.
For two sections s and R, as above, we prove tha t
LEMMA 2.1. For any smooth vector field X over U, the following formula holds:
K o Tbs. Xb -- K o T(R(b),~(R(b))) ~ • [,(R(b), (R*o~),,. Xb), T~(a o R ) . X~]
for every b E U.
PROOF. By differentiat ion of (2.4) we get
(2.5) Tbs(Xb) -- T(R(b),~,(RO)))~" (T~R(Xb), T~(ao R) . (Xb)).
2 9 4 VASSILIOU: TRANSFORMATIONS OF LINEAR CONNECTIONS
I f we decompose ToR(Xb) into its vertical and horizontal components, using (2.1) we easily check tha t
(2.6) Tb_R(Xo) = ~(R(b), (R*eo)b. Xb) + c(R(b), Xb).
In virtue of (2.3) and (2.6), we transform (2.5) into
Tbs(Xb) -~ T(R(b),¢(R(b))) ~ • [v(R(b), (R*eo)b" Xa), Tb((r o R) . Xb] + 7(s(b), Xb).
Since Ker (K) ---- Im (~), the last equality implies the formula of the lemma. |
For a given local chart (U, ~) of B let us denote by 8, R, X the principal parts (instead of s, etc.), in order to simplify some formulas below. Then we have
PROPOSITION 2.2. Over a local chart (U, iv), the [ollowincj [ormula holds:
l~,(x)(~, X'x) -- [R(x) o (~*e%)~ • (X~) o R(x) -~ -- (DR(x) . Xx) o R(x)-~] . (-~),
where x E IV(U), ~p :=-~-1 and m~ := R*eo.
PROOF. This is the local form of the formula of Lemma 2.1. Briefly, it is deduced as follows:
First observe that, if x = ~(b), then X b = Tx~(Xx) and (R*eo)b(Xb) ----- ----- [(R o ~)*eo]x(Xx). Then, by means of t h e local structure of our bundles, we
t idF) is the morphism check that , 1 o e a 1 1 y, T(R(b),e) ~ (here e =
L(F) --,- B×L(F) :g ---, (0, R(x) og);
the morphism T(R(b),e(R(O)))~ is given by
BxL(F) X Y --,- BxL(F) : (y, g, v) --+ (y, g o (a o R o V')" x + R(x). v).
Finally, Tb(a o R ) . X b is locally identified with D(a o R o ~2)(x).X x. On the other hand, differentiation of the local expression of (2.4) implies
that
D~(x) . X--" x = DR(x) . Xx o (o o R o ~0)(x) + R(x) o D(a o R o ~2)(x) • X~x.
The formula of the Proposition follows now, after some easy (although lengthy) calculations. II
We can write the formula of the previous Proposition under the follow- ing useful form:
(2.7) r Jx ) (u , y) = [R(x) o (~*~)x- y o n(x) -1 - DR(x). y o R(x) - l ] (u),
for every y E B and u E F. This is the case, since, for every x = iv(b), we can determine X and 8 such tha t X(x) ----- y and 8x = u.
In particular we have
VASSILIOU: TI~ANSFOla~ATIONS OF LINEAR CONNECTIONS 2 ~ 5
1~). COROLLARY 2.3. Let R be the local section of l(E) given by R(b) :-~ l~(q~(b), Then
r,(x)(u, y) = [(~,%)~. y](u),
]or every x ~ ~(U), y ~ B and u ~ F.
PROOF. This is an immediate consequence of (2.7), since the assumption implies tha t R(x) ---- 1 F, for every x ~ ~(U). |
For the sake of completeness, we shall show how the preceding Corollary leads to the classical formulas of the finite-dimensional theory. We assume that B is m-dimensional and F ---- R". For a chart (U, ¢) of B we denote by x i, i ~ 1 . . . . . m, the corresponding local coordinates and by X~, i ~ 1 , . . . , m, the base field over (U, ~). Also, we denote by (e~),= 1 ..... n the natural basis of R" and we define the local sections ~ : U - ~ E given by ~(b) : ~ ~-l(e~),
~ 1 . . . . , n. In this case we may set ~ ~ (~1 . . . . , ~n). Finally, we denote by (E~),.~ the basis of gl(n, R ) ~ - L ( R " ) (: the Lie algebra of GL(n, R)). With respect to the latter, the local connection form e% :---- R*eo is expressed by ~ , ~ ~ , ~ co~.E~. Each entry o~ is an R-valued form on U such that
( o ~ , ( x ) ) ~ . ( ~ ) = Z ~ ( o ~ ( x ) ) ~ . ~p,
for every b E U and any vector field X over U. The finite-dimensional Christof- fel symbols are now smooth maps of the form r/~ : u -~ R, so that
(2.8)
(of. also [5]; p. 367),
b E U .
C o m : ) ~ Y 2.4. The following ]ormulas hold over U:
(i) [%(x,)](e.) = ~,~r,"~.e~,
(~) vx, e. = ~,~r,~. ~ ,
(~i) v×~ = ~ , ~ ( x ) . ~;
(iv) eo~ = ~ . . p.,(rf~ • d ~ ) ~ .
X ~ ~ (B),
PROOF. The first equali ty is a direct consequence of Coroll~ry 2.3 and t h e
previous definition, if we set u -~ e~ and y = e t. For the second equality, we observe that the definition of the eovariant
derivative (of. [6]; p. 3) yields the following local expression for the principal pans
296 .V~SSI~IO~:, TRANSFORMATIONS OF. LINEAR.' CONNECTIONS
Since now ~i(x.) =.e~ and Xt(x ) --i e i, for every x E ~0(U), the above formula and the definition of principal parts (el, Section 1) imply that
thus getting (ii). Equali ty (iii) is an immediate consequence of (i) ( : r~ = o ~ ( x i ) ) and
the properties of covariant derivatives. Finally, since every vector field X ' o v e r U has the expression X - :
~- , ~ i d x t ( X ) . X i , we check that
) -
Comparing the last equality with the defiriition of the components of o~¢, we conclude fhat
(~( / ) )~ = r~(b). ~t(dx ' ( i ) )b , ¢
which implies that
(o)~(X)) b -- ~'=,~(eo~(X)) b • E~ -- 2:=,~(r~=(b). Z~(dx~(i))b) • E~ -
= Z,,~,p(r~dxt(X))b. E~.
The last equality concludes the proof of (iv) as well as the Corollary. |
NOTES. Formula (ii) of the previous Corollary is also proved by [5]; p. 370, in the finite-dim'ensional context.
I f E - - T B , then the sections { ~} coincide with (Xi); hence, the for- mulas of the previous Corollary, properly modified, yield the classical formulas found in standard references (cf. e.g. [4]; pp. 264 265 and [8]; pp'. 140 143).
§ 3. Related Co.nneetions :
• Let 1 ~ (P, G, B, ~) and l' -- (P', G', B', ~') be two principal fibre bundles provided with the connections e0 an d o)' respectively. Assume that ' (f, ~, h) is a p.b-morphism of I into l'.
DE~NITION 3.1. Two connections co and co', a s above, are said to be (f; ~, h)-related if thecorresponding splitting morphisms c and c' satisfy the equality c' o (f ~( T h ) ~ T.f o c. . . . . . .
Hence, roughly speaking, the given morphism maps the horizontal subspaces of c in to the horizontal subspaces of c' . W e get the following com-
V ' A S S I L I O U : T R A N S F O R M A T I O N S O F L I N E A R C O N N E C T I O N S 297
mutat ive diagram of split exact sequences
: " . v T n l
0 " P X G ~ T P 7 - - -+ ~*(TB) " 0
IX • f X T h
i ' 7 - - - -*u '* (TB ' ) , 0 V" c"
with ~ denoting the L i e ' B a n a c h algebra homomorphism induced by ?. .If a : I -~ B is a smooth curve of B, the parallel displacement ~ along
is defined as in the classical e a se .
PROPOSITION 3.2. Under the above notations, the [ollowing conditions are equivalent:
(i) c' o ( f×Th) : T / o c,
(ii) V" o T / : ([X~) o V,
(iii) T/(u v) : (T/(u~')) ;
(iv) T/(u h) = (Tf(uh'));
(,~) I*~o' = ~ . o~,
(vi) I o ~ = ~:, o I I ~ -~ (~ (0 ) ) ,
/or every smooth curve ~ o/ B with a' := h o ~.
u C T P .
u 6 T P .
PEOOF. The equiValence of the first five conditions is given in [17]. The equivalence of the last condition with all the others can be found in [19]. |
Let now 1 and l' be two principal bundles with the same base B and the same structural group G. We fix an open covering ~---- {Ui}i~ I and a family O f lpcal sections :{st} ~ ~ z o.f!, over ~. We denote by gq the corresponding transition functions of 1. Similarly, for the same covering, we fix the family of local sections {s~}~.Ex of l', and we denote by g~i the corresponding transition functions.
L~MMA 3. 3 ([2]; n o 6.414). For every G--B:isamorPhism ([, id c, ids) of 'l onto l', there exists a unique family o f smooth morphisms h,:U i ~ G: (i E I) such that
g;j(x) = h~(x) .g~j (x) , hi(x)-1; x 6 Ut N Uj,
for every pair of indices i, ]. Moreover, th e following:, equality
f ( s , ( x ) ) = ~;(x) . h,(x) ; . . . . x ~ V,, • . . ' . " ' , , . i . , '
holds for every i E I . . . . : . . . . .... :
2 9 8 VAS$ILIOU: TRANSFORMATIONS OF LINEAR C O - - I O N S
Using the previous Lemma, we can prove the nex t result (cf. [17]), which is a criterion based on local connection forms. I t will be used as a basic tool below.
TH~,OI~.M 3.4. Let 1 and l' be two ~arincipal bundles with the same base B and the same, structural gron~ G. 1[ eo and co' are two connections on 1 and l' respectively, then the following conditions are equivalent:
(i) There exists a G--B-isomorphlgm ([, idG, idB) o[ 1 onto l" so that eo and co' are related.
(ii) For a fixed covering ~ o[ B, there exists a [araibj o[ smooth morphisms {hi}ie l, as in Lemma 3.3, so that the corresponding local connectio~ forms satisfy the condition
~oi = Ad(h~1)o~i + hi -1 "dhi
for every i E I .
The proof is a bit technical and it is based on the construction of a con- nection form via the local connection forms. Note tha t the above condition is the symbolic expression of the 1-form given by (~t)b(u)= Ad(hl(b)-t ) (0~;)b(~) + Tb(t~(b)-, o hl)(u ), where b E Ui, u E TbB and 2g (g E (7) denotes, in general, the left translations of (7.
For vector bundles and (linear) connections we have the following analogue of Definition 3.1:
DEF~mTION 3.5. Let L = (E, B, p) and L ' = (E', B ' , Is') be two vector bundles with corresponding connections 7 and 7'. I f iF , h) is a v .b-morphism of L into L ' , then 7 and 7' are said to be (F, h)-reh~ed i f
7" o (FxTh) = TF o 7.
As in Definition 3.1, we have an analogous commuta t ive diagram and, if we denote by T~ the parallel disp~cement of ~ (with respect to 7), we get the following analogue of Proposit ion 3.2.
PROPOSITION 3.6 ([12]; p. 41). Let 7 and 7' be two linear connedions on L and L ' r ~ e a i v e l y . For a v.b-morphism (F, h), as above, the [ollowing conditions are equivalent:
(i) TF o 7 = ~" o (FXTh).
(ii) F o K = K' o T,F.
(iii) For every 8~nooth section s : I -,. E of L along ~: I -,. B, the following
e ~ t a y holds
VASSILIOU: TRANSFORMATIONS OF LINEAR CONNECTIONS 299
F o V~s = F ' , ( F o s),
where o~' -~ h o a.
(iv) For every smooth curve a : I ~ B.
F o T~ = T ' . o F I P-I(O) •
We note tha t if the connections are not linear, we obtain only the equi- valences (i)¢~(ii)~(iv).
Using now the notations introduced in the previous sections, we have the following useful local criterion
PROPOSITION 3.7. Two linear connections ~, ( ~ K ) and ~,' ( ~ K ' ) , on J~ and L' respectively, are (F, h)-related if and only if, over every pair of suitable charts (U, q~) and (U' ~d), the corresponding Christoffel symbols satisfy the fol- lowing equality:
(F~)~[r~(x)(u, y)] ---- r;,(h~(x))[(F~)~(u), n(h~)(x), u] + D(F~)U(x).y. u,
for every x E ~o(U), u E F, y E R.
PROOF. In virtue of Proposition 3.6, it suffices to find the local expres- sion of Condition (ii). Indeed, for every (x, u, y, u') E T ( U ) X F X B × F , we have tha t
[@' o (F o K) o T@-I] (x, u, y, u') : [(@' o iv o @-1) o K~] (x, u, y, u') - -
- (~,' o F o ~ , -1) (x, r~(x)(u, y) + u') - (h~(x), (F~)~(r~(x)(u, y) + u')).
Similarly,
[~ ' o (K' o TF) o T@-I] (x, u, y, u') : (K~, o TF~)(x, u, y, u')
: K~o[(F~)(x, u), D(F~)(x, u). (y, u')] --
-~ K~,[h~(x), (F~)~(u), D(h~)(z).y, D(F~)*t(x).y. u + (F~)~(x).u'] --
----- [h~(x), P~,(h~(x))((F~)~x(u), D(h~)(x). y) + D(F~)**(x).y. u + (F~)U(x). u '] .
Our assertion follows immediately from the above equalities. |
In particular, we have the following consequence of Proposition 3.7, which will be used frequently below
COROLLARr 3.8. I f L and L ' have the same base B and (iv, ids) is a v.b- isomorphism of L onto L' , then K and K ' are (iv, ids)-related i f and only if, as in the previous Proposition,
(F~)~(r~(x)(u, y)) =- r;(x)((F~)~(~), y) + D(F~)~(x).y.~, for every (x, y , u) E q~(U)× B X F and for each local cha~rt.
3 Periodica Math. 18 (4)
3 0 0 VASSILIOU: TRANSFORMATIONS OF LINEAR CONNECTIONS
NOTE. Related connections, considered in this paragraph, can be called conjugate, as an extension of the classical notion due to A. P. ~ORDEI~ (of. e.g. [7], [11] and their references). We also note that related connections on vector bundles are called, by [12], adaptges.
§ 4. T r a n s f o r m a t i o n s o f c o n n e c t i o n s
In this section we shall relate linear connections, defined on vector bun- dies, with their corresponding (infinitesimal) connections on the bundles of frames.
DEF~rrIoN 4.1. Let L = (E, B, p) (resp. L ' ~- (E', B ' , p ' ) ) be a vector bundle of fibre type F (resp. F') and let ~ be a Lie group homomorphism of GL(F) into GL(F'). A continuous linear map 1 EL(F, F') is said to be com- patible with q~ if l(g(u)) --- ~(g)(l(u)), for every u E F and g E GL(F).
The following Lemma is a particular ease of [2]; n o 6.5.5, stated without proof (cf. also an analogous result in [3]; Problem 9, p. 101).
L~YrMA 4.2. Let (/, q~, h) be a p.b-morphism o/ l(E) into l(E') and let l EL(F, F') be any continuous linear mal~ compatible with qD. ~hen, there exists a unique v.b-morphism (F, h) of L into L' such that
F o e = e ' o if×l).
PROOF. Let e be an arbitrarily chosen element of E. Then e ---- ~(r b, u) - -
---- r(u), with u E F and r b ~- (b, r), r E Lis(F, Eb). We define F by:
F(e) : = d(t(~b), l(u)).
I t is easy to cheek tha t the above definition is independent of the choice of the representative (r b, u) giving e. Thus, we get immediately the desired formula. The same formula, along with the surjectivity of e, implies that F is a smooth map. Next, in order to show that (F, h) is a v.b-morphism, we have to verify conditions (LVBM 1 ) - (LVBM 4) of [1]; p. 12. In this case, we see that (F~) u (cf. Section 1) can be given by
(F~)~(x) = H~(b) o r ' o l,
where b : = vj-!(x ) E U and (h(b), r') : = f(rb). Notice that in the formulas (1.4), (1.5) we set the subscript y in order to avoid confusion with the Lie group homomorphism ~. Hence, we have that
(_F,)(x, u) = ((h~ (x), (F~)~(x). u).
VASS]LIOU! TRANSFORMATIONS OF ]SINEAR CO~STECTIO57S 301
Since F is a smooth map, so are F~ and (F~)U; thus , the above mentioned conditions are indeed satisfied. The uniqueness of F is obvious and the proof is now complete. |
THEOREM 4.3. Let JL and L ' be two vector bundles endowed with the linear connections K and K ' respectively. Assume that (/, % h) is a p.b-morphism of l(E) into l(E') and 1 EL(F, F') is a map 1 compatible with q~. I / (F, h ) i s , t h e unique v.b-morphism defined in Lemma 4.2, then the following conditions are equivalent:
(i) K and K ' are (F, h)-related. (ii) The corresponding infinitesimal connections eo and co' are (], ~, h)-
related.
PROOF. In virtue of Lemma 4.2 and equality (2.3), for every (e, v) E E × s T B with e : ~(r b, u) : r(u), we have that
(4.1) [•' o (FxTh)] (e , v) : 7'(F(e), Th(v)) -- [T~/(r,),,(u))~' o c'] (/(rb), Th(v)). By differentiation of the formula of Lemma 4.2 we get
(TeF o T~r,,,) e) (c(rb, v)) -- [TI/(,,),,(,)) e' o TrJ] (c(rb, v)).
Hence, if (i) holds, then, in virtue of Proposition 3.6, equality (4.1) and the last equality above, we conclude tha t
TI/(r,),,(u)) e'[c' o (/X Th) (r b, v)] : Tb(~,),,(,)) e'[(Tl o c)(rb, v)]. I t is easy to show that the restriction of T~, on each horizontal subspace is an injection (eft also [ 18 ]); thus, in virtue of Proposition 3.2, we get Condition (ii).
The converse follows easily, using the same equalities, as above, in the opposite direction. |
For later reference we state the following immediate consequence of the preceding result, namely we have
COROTJTJA~Y 4.4. Let f :-~ ,(f, id~L(F ), /ds) be a GL(F)-B-automorphism of l(E). As~ame that K and K ' are two linear connections on L with corresponding infinitesimal connections o~ and co" (on l(E)). I f F : : (F, ids) is the v.b-auto- mor,lahism o / L , defined by f and 1 : = id T (cf. Lemma 4.2), then the following conditions are equivalent:
(i) K and K ' are F-related, (ii) o and co' are frelated.
1 F and F' are standard fibres of L and L', respectively.
3*
~ 0 9, VASSILIOU: TRANSFORMATIONS OF LINEAR CONNECTIONS
We can partially reverse Lemma 4.2 in the following way:
L ~ 4.5. Each v.b-automorl~hism (F, ids) of L induces a GL(F)-B- automorphism (~, id~L(~), ids) of l(E). Moreover, we have that
Fo e= eo (/Xid~).
PROOF. We define the map ~ : P ( E ) - ~ P(E) with ] ( rb) := (b, F~or), where r a = (b, r) and F b is the restriction of F on the fibre E b. The differen- tiability of ~ is checked locally, as follows: for a chart (U, ~) of B and a suitable chart of l(E) (ef. Section 1), the local expression of ~ is L,o~oL-~:q~(U)X xLi s (F ) -~ ~(U)xLis(F) . Then, for every ix, l )6~(U)×Lis (F) , there exists rb E P(E) such tha t L~(rb) = (x, l) ---- (q~(x), ~b o r). Hence, by the above de- finition and (1.1), we have that
(L, o fo L~)(~, l) = (~, (~b 0 E~ o ~) o l) = (X, (~ o F o ~-~)~ o 0" Since F is a v.b-morphism (~ o F o ~-I)~ is a smooth map and this implies the differentiability of ~.
I t is trivial to check now that the triple ([, idGu~}, ids) is a GL(F)-B- automorphism of l(E} satisfying the equality of the statement. |
The following result may be thought of as an inverse form of Corol- lary 4.4.
THEORE~ 4.6. Let F and ~ be as in the previous Lemma. Then two Hnear connections K and K' or, L are F-relate~l if and only if the corresponding con- nections co and a/ on l(E) are ~-relafexl.
PROOF. We shall prove our assertion using the local machinery provided by Theorem 3.4 and Corollary 3.8. For this purpose, we consider an open covering ~ of B given by a family of local charts (U i, ~t), i E 1, and we fix a family of local sections Cover ~) {Ri}t~ i of l(E). As we explained in Lemma ~ 3.3, ~ is completely determined by a properly defined family of smooth mor- phisms h~ : U i --~ GL(F), so that
(4.2) f(Ri(b)) = Rt(b). hi(b),
for every b 6 Ui and every index i 6 I. I f we denote now by •i the principal part of R i (el. the notations of Section 1 and the commen~s preceding Propo- sition 2.2), working as in the proof of Lemma 4.5 we have that (omitting double subscripts)
(L~o/oR~)(b)--(L~o]oL~1)o(L~oR~).(b)_~ ,
-- (L~ o / o L~1(Ti(b), Ri(~t(b))) -- (~i(b), (~ o F o @-~)~(b) o Ri(~(b))).
VASSILIOU: TRANSFOI~MATION8 OF LINEAR CONNECTIONS 303
Sett ing x :----~i(b), the last equal i ty combined with (1.3) and (1.4) implies t h a t
(4.3) (pr 2 o L~)( f (R, (b)) ) = (F~)~ o R~(x),
for every b E Ur Similarly, we check tha t
l~(Ri(b) . h,(b)) - : l ,[(b, (pr 2 o R, ) (b) ) . hi(b)] :
: 1,(b, p r 2 (R,(b)) o h,(b)) : = (b, ~ b o pr2(R,(b)) o hi(b)).
Thus, by the definit ion of the action locally,
(4.4) (pr 2 o l~)(R~(b), hi(b)) = R~(x) o h~(q~ll(x)).
Since p r 2 o L~ = p r 2 o l~, equalities (4.2) (4.3) and (4.4) imply tha t
(4.5) (Fv)~ o (R~-(x))) = R,.(x) o ((h~- o ~i-1)(x)),
for every x E 7~i(Ui). On the other hand, by differentiat ion of (4.5), for every x E ~0i(Ut) and y E B (: tile ambient space of the chart), we have tha t
( f ¢ ) ~ o ( D R , ( x ) . y ) + (D(F¢)U(x ) . y ) o R~(z) = (4.6)
R, (x ) o (D(hi o ~7-1)(x) . y ) + ( D R i ( x ) . y ) o ((h+ o +7')(x)).
Also, we observe tha t h i o ~-1 : T i ( U i ) + GL(F) gives now
T ( h i 0 ~T1)(X ,y) ~-- ((h i o cfT1)(x), D(hi o cf-i-1)(x) . y ) ;
thus, ff we set y : ~ T~cfi(v ), for a vector v ~ T b B ,
(4.7) Tbhi(v ) -~ D(h i o q~-l)(x), y.
Then, applying (4.5) and (4.7) in (4.6), we conclude tha t
(4.8) f D(F~)U(x) "y = R , (x ) o Tbh~(v ) o R ~ ( x ) - I + (DRt ( x ) . y ) o hi(b ) 0 R i ( X ) --1
, - - R i (x ) o hi(b) o i~ i (x) -1 o ( D R i ( x ) • y ) o R i (x ) -1.
The previous pre l iminary discussion proves the theorem as follows: I f K and K " are F-re la ted, then Corollary 3.8 holds; hence, in vi r tue of the general formula (2.8) (where now we set e% ---- oJt), combined with equalities (4,5) and (4.8), we check t h a t (recall also t h a t v: :---- ¢~-1):
RI(X ) o hi(b ) o Op*coi)x(y ) o R t ( x ) - x _~
= R t ( x ) o Tbh~(v) o R~(x) -~ + R~(x) o (~p*~ol)~(y) o hi(b) o R~(x ) - x
or, since R~(x) ~ GL(F), x ---- ¢~(b) and y ~- T v ~ ( v ) ,
(eoi)a(v) = h~(b) -~ o (col)a(v) o hi(b) + h~(b)-I o T~hi(v),
t h a t is
(4.9) (eo~)b(v) ~- Ad(h~(b)-~)(eo~)a(v) + (hi - ~ • dhi)~(v),
3 0 4 V:ASSILIOU: TRANSFORMATIONS OF LINEAR CONNECTIONS
for every b~ U s and v E TbB~ Thus, in virtue of Theorem 3!3, we conclude tha t oJ and o~' are f-related.
Conversely, assume t h a t o~ and o~' are f-related. Then, (4.9) and the other equalities as above imply that the formula of Corollary 3.8 holds; hence, K and K ' are F-related. II
§ 5. Applications
A. Affine transformations
Using the machinery of' Section 4, we can extend the classical notion of affine transformations within the general context o f arbitrary vector bundles. More precisely, we fix a vector bund le / ) with a linear connection K. Then,
DEFINITIOI~ 5.1. An automorphism F : ~ (F , /ds ) of L is said to be an affine transformation of L (with respect to K) if it commutes with the parallel displacement along every smooth curve ~ of B , i.e.
F o T , = o F I
I f we denote, again, by to the infinitesimal connection on l(E), corre- sponding to K, then we give also the following
DEFIN1TIOI~ 5.2. A GL(F)-B-automorphism f of l(E) is said to be con- nection preserv~n(1 (with respect to ~o) if f*~o = co.
L~ms.~ 5.3. A connection preserving automorphism of l(E) determines a unfflu~ affirte transformation of L.
PROOF. Let f : = (f,/daL(~,/ds) be a connection preserving automor- phism of l(E). Then, f a n d 1 : = idr determine a unique v.b-automorphism (F,/ds) of L, in virtue of Lemma 4.2. Our assertion follows now from Corollary 4.4 and Proposition 3.6, properly modified. II
Conversely, we have
L ~ a _ ~ 5.4. Every afflne transformatiort of L ir~uees a unQue connection preserving automorphism of l(E).
PROOF. The Lemma is an immediate consequence of Proposition 3.6 and Theorem 4.6. |
V A S S I L I O U : T R A N S F O R M A T I O N S OF L I N E A R CONNECTIONS 305
THEOREM 5.5. An automorphism of l(E) is connection preserving if and only if it is induced by an affine transformation of L.
PROOF. Assume first that f is a connection preserving automorphism of l(E). Then, in virtue of Lemma 5.3, the v.b-automorphism F of L, determined by f and id r, is an affine transformation. On the other hand, ~' induces a connection preserving automorphism ~ of l(E), in virtue of Lemma 5.4. We shall show that f ~ 7. Indeed, let r b ~ (b, r) be any element of P(E) and let us set f(rb) := (b, r'). Following the construction of Lemma 4.5, we see that
A (5.1) [(rb) = (b, F b o r).
Also, for every u E F, in virtue of Lemma 4.2, we check that
r '(u) = eff(rb), u) = F(e(rb, u)) = Fff(u)) = (F b o r)(u);
hence, r ' ---- ~'b o r. The previous equality, along with (5.1), proves the first part of the Theorem.
Conversely, if f is induced by an affine transformation, then f is con- nection preserving, as a consequence of Lemma 5.4. Thus, the proof is c o m -
p|ete. I
NOTE. The previous Theorem is the infinite-dimensional v.b-version of the classical result given in [8]; p. 228. In the latter case the tangent bundle of a finite-dimensional manifold is used instead.
B. Ordinary linear equations
Let L ~ ( R X F , R, prl) be the trivial vector bundle over R of fibre type a Banach space F. We assume that K is a linear connection on L. We shall interpret K as an ordinary linear equation in Banach spaces and the transformations of (related) connections as the ordinary transformations of (equivalent) equations.
With respect to the usual (countable) atlas of R, we consider the Christof- fel symbols of K, that is the smooth maps
rl : q~i(Ul) -+ I2(FX R, F).
In paxtieular, we denote b y r 1 the symbol defined over the chart (R, ida). Then we set
(5.2) Ai(t ) : = l~z.(t)(.,/); t E q~i(U~),
where I is the unit of R. In particuJar, we set A--- - -AI: /~-+ L(F). I f A i and A 1 correspond to
charts (Uz. , 9i) and (U i, q~i) with U i N U i ~= fJ, then the compatibility condi-
3 0 6 VASSILIOU: TRANSFORMATIONS OF LINEAP~ CONNECTIONS
tion of the Christoffel symbols (cf. e.g. [6]; p. 5) implies that
(5.3) . 4 j ( t ) = o . o
for every t E ~j(Ui n ui), where the upper dot denotes the usual derivative of maps over R. In particular, for i = 1, (5.3) yields
(5.4) Aj(t) ----- (~)-l)(t). A(qgfl(t)); t E q~j(Uj).
I~OPOSITION 5.6. Each linear connection K of L corresponds, in a bijective way, to an ordinary differential equation dx/dt = A(t) .x, where [A(t)](u)= ----- --rl(t)(u, _/), for every u E F and t E R. Moreover, the solutions of the equation determine the horizontal global sections of the given trivial bundle.
PROOF. For a given connection K, we define the corresponding equation as in the discussion preceding the present statement. Conversely, for a given equation with coefficient A : R -* L(F), we define the smooth maps .41 : qJi(Ui) - , L(F) by (5.4). The sam e equality proves that .4t and A 1 satisfy the compati- bility condition (5.3). Then, we define the Christoffel symbols {I'i}l~ 1 with ri(t)(u, s) = - 8 . [.4i(t) ](u), for every t E q~l(Ul), s E R and u E F. In virtue of (5.3), the above symbols satisfy the ordinary compatibility condition over overlapping charts; hence, we may define a linear connection K on L, having as Christoffel sumbo]s the previously defined family (cf. [6]; p. 2).
I f now ~ : R - . F is a global section of L, we can write ~(t) = (t, ~(t)), where ~ : R -* F. The assumption that ~ is horizontal implies that K(~(t)) = 0, for every t E R. Thus, in virtue of (2.2) and the previous notations, we conclude that
~(t) = [.4(t)](~'(t)); t ~ R,
from which follows t h e last assertion of the Proposition. |
NOTE. The minus sign in the definition of A, v ia / ' t , is only for technical reasons, as it is clear from the proof of the previous Proposition.
THEOREM 5.7. Two linear connections K and K ' on L are (F, idR)-related if and only if the corresponding equations dx/dt ---- A( t ) .x and dy/dt : B( t ) .y are equivalent, i.e., there exists a smooth transformation Q : R --* Lis(F) such that x----Q(t).y, or, equivalently,
(5.5) B(t ) = Q ( t ) - i o (A( t ) o Q(t) - Q((t)), for every t E R.
PROOF. I f the connections are related, then Corollary 3.8 and the above definition imply that
VASSILIOU: TRANSFORMATIONS OF LINEAR CONNECTIONS 307
(F1)~(A(t) • u) -~ B(t)((F1)~(u)) -- ((F~)u)'(t) . u
for every t E R and u E F. Setting (F1) ~ -~ Q(t)- l , we conclude that (5.5) holds. Conversely, if the equations are equivalent, we define the map
F: R × F ~ R × F : (t, u) --. (t,Q(t) • u).
We easily check that (F, ida) is a v.b-automorphism of Z. Moreover, (5.5) implies that the condition of Corollary 3.8 is satisfied; hence, the two connec- tions are related. |
To conclude the note let us assume that F is finite-dimensional, say, F -~ Rn. Then, as we have seen in the discussion preceding Corollary 2.4, the finite-dimensional Christoffcl symbols of a connection K, for the chart (R, idR), are given by
~ r ~ ( t ) • e~ ---- rl(t)(e ~,/_); a, fl = 1 . . . . . n.
Thus, if (A~(t)) is the matrix corresponding to the coefficient A, where [A(t)](e~) -~ , ~ A ~ ( t ) . e ~ , we see that
(5.0) = A~(t); t E R.
Hence, we can prove the following Floquet-type result on connections
COROLLARY 5.8. IT/K is a linear connection with periodic coefficient A ~--- (A~) then there exists a linear connection I¢." with constant Christoffel symbols over R and such that K and K ' are E-related, via an automorphism (F, idR).
PROOF. In virtue of the classical theorem of G. FLOQUET (cf. [14]; p. 146, [16]; p. 24), the differential equation with coefficient A(t) : (A~(t))~) is equivalent to an equation with constant coefficient B---- (B~). Hence, by Theorem 5.7, there exists a proper automorhpism F :=- iF,/dR) of the bundle such that K and K ' (: the connection corresponding to B) are E-related. Furthermore, the Christoffel symbols of K ' , over R, are given as in (5.6), tha t is I'll(t) ---- B~, for every t E R. The proof is now complete. I
R E M A R K S .
(i) A similar result, a s bore, can be obtained in the infinite-dimensional case. However, since such an extension relies on the infinite-dimensional version of Floquet 's theorem (cf. [10]), certain restrictions on A and F should be imposed.
(ii) The periodicity of A may give rise ~o a more general s tudy of fiat connections on certain vector bundles of the form (E, S 1, p), where S 1 is the unit circle. The idea to work on such bundles, in order to obtain the classical Floquet 's theorem, is given by R. G~RARD--G. REEB ( A n n . ~c.
308 VASSILIOU: TRANSFORMATIONS OF LINEAR CONNECTIONS
N o r m . S u p . iP i sa 21 (1967), 9 3 - - 9 8 ) . H o w e v e r , t h i s i d e a n o t o n l y y i e l d s a
g e o m e t r i c p r o o f o f F l o q u e t ' s t h e o r e m , b u t a l so , c o m b i n e d w i t h t h e m e t h o d s
o f t h i s p a p e r , l e a d s t o s o m e F l o q u e t - t y p e t h e o r e m s o n t h e c o n n e c t i o n s o f a
b u n d l e o f t h e p r e v i o u s t y p e .
R E F E R E N C E S
[1] R. ABRAHA~ and J. ROBBIN, Transversal mappings and flows, Benjamin, New York, 1967. M R 39 ~ 2181
[2] 1~. BOURBA~, Varidtds di//drentielles et analytiques; ~asc~cule de rdsul~ats §§ 1--7, Hermann, Paris, 1967. M R 36 4~ 2161
[3] J . DmuDo~-~ , Treatise on analysis, 111, Academic Press, New York, 1972. M R 5O 4# 3261
[4] J . DIEUD0~-~, Treatise on analysis, I V , Academic Press, New York, 1974. M R 50 4# 14508
[5] T. V. Duc, Sur la g~om~trie diff~rentielle des fibr4s vectoriels, KSdai Math •em. Rep. 26 (1975), 349--408. M R 52 ~ 9111
[6] P. F~scm~T. and W. KLn~GEI~BERG, Riemannsche Hilbertmannig]altigkeiten. Perlo. dische geoddtische, Lecture Notes 282, Springer, 1972. M R 49 4# 6275
[7] J. GANCARZEWmZ, Connexions conjugu~es, Colloq. Math. 26 (1972), 193--206. M R 48 4# 12383
[8] S. KOBAYASHI and K. N o ~ z u , Foundations o] di]]erential geometry, I , Interscience, New York, 1963. M R 27 4# 2945
[9] S. LA~G, Di]]erential mani/olcls, Addison-Wesley, 1972. M R 55 @ 4241 • 0
[10] J. L. MASSERA and J . J . SCHAEFFER, Linear di]]erential equations and ]unct~on spaceS, Academic Press, l~ew York, 1966. M R 85 ~ 3197
[11 ] D. OPl~IS, Sur les connexions infinitdsimales conjugu~es d 'un espace fibr~ principal diff~rentiable, An. Univ. Timi~oara, Set. ~ti. Mat. 11 (1973), 75--89. M R 50
8357 [12] J. P. PE~OT, De submersions en ]ibrations: Expos~ du s~minaire de g6om~trie diff~-
rentiellc de Mlle P. Libermann, Paris (1967) [13] J. P. PENOT, Connexion lin~aire d~duite d 'une famille de cormexions lin~aires par
un foncteur vectoriel multilin6aire, C. R. Acad. Sc. Paris 262 (1969), A. 100-- 104. M R 39 ~¢ 2182
[14] L. S. PO~-TRYAGr~, Ordinary di]/erential equations, Addison-Wesley, 1962. M R ~1 @ 3650
[15] P. M. Q u ~ , Introduction d la gdomdtrie des vari~tds di//drentiables, Dunod, Paris, 1969. M R 39 ~¢ 3415
[ i6] M. ROSEAU, Vibration non lindaires et thdorie de la stabilitd, Tracts in Na tura l Phil- osophy 8, Springer, Berlin, 1966• M R 38 4# 5171
[17]E. VASS~LIOU, (], ~, h)-related connections and Liapunoff 's theorem, Rend. Circ. Mat. Palermo 27 (1978), 337--346. M R 81d: 58013
[18] E. VASSI~IOU, Affine transformations of Banachable bundles of frames, Math. Balcanica 6 (1976), 291--295. M R 81c: 58012
[19] E. VASSIHOU, On affine transformations of Banachable bundles, Colloq. Math. 44 (1981), 117--123.
[20] J. VILMS, Connections on tangent bundles, J. Dill. Geometry 1 (1967), 235--243. M R 37 @ 4742
(Received January 2!, 1 9 8 0 )
UNIVERSITY OF ATHENS IBSTITUTE OF MATHEMATICS 57 SOLONOS STR. ATHENS 143 GREECE
