Yano-ledger connection and induced connection on vector bundles

Acta Math. Hung. 59 (3-4) (1992), 405-421.

YANO-LEDGER CONNECTION AND INDUCED C O N N E C T I O N ON V E C T O R BUNDLES Z. KOV]tCS (Nylregyh~za) and L. TAM/kSSY (Debrecen)

Introduction

K. Yano and A. J. Ledger [13] constructed from a linear connection V on a manifold B a torsion-free linear connection on T B (called the Yano-Ledger connection). M. M a t s u m o t o [7] proved that: a) V determines a Finsler connection (75, V) in the space V T B of Finsler vectors; b) the symmetrization of the extension V ~ of (75, V) to T T B is exactly the Yano-Ledger connection on TB; c) the Levi-Civita connection of the Riemannian metric on T B derived from a R i e m a n n i a n metric g on B coincides with the Yano-Ledger connection derived from the Levi-Civita connection V of g iff the Riemannian curvature tensor of g vanishes (see also [2]). On the other h a n d R. Miron [8] developed a theory of Finsler connections on vector bundles. T h e purpose of this paper is to construct a Yano-Ledger connection on vector bundles, and t h e n to prove the analogues of M a t s u m o t o ' s theorems a), b), and c) for vector bundles and vector bundle Finsler connections. In our considerations and constructions we apply pullback of pseudoconnections. T h e y are developed and investigated in §§1, 2, and 3. §4 yields the Yano-Ledger connection for vector bundles, and §§4, 5, and 6 present the mentioned theorems analogous to those of M a t s u m o t o . Concerning n o t a t i o n and terminology we refer to the m o n o g r a p h s [1],

§1. P r e l i m i n a r i e s A) P s e u d o c o n n e c t i o n s . This notion is a generalization of the linear connection. Pseudoconnection was introduced by Y. Wong [12]. In this paper pseudoconnection is defined over a pair of vector bundles ~ = (E, 7r, B, V r) and ~ = ( E , ~ ' , B , V 8) with c o m m o n base space B. E denotes the total space, 7r : E --* B is the projection, and V r is a real vector space of rank r. A pseudoconnection in ~ w.r.t. (with respect to) ~ is a pair ( V , A ) of maps. A : ~ ~ rB is a strong vector bundle mapping, i.e. such t h a t the fiber ~ - l ( x ) , x E B is linearly m a p p e d into the tangent space TxB. V : Sec ~ x Sec ~ ~ Sec ~, (5, r/) ~ ~ 7 ~ is additive in b o t h variables 5 E Sec

406

z. KOVACS, L. TAMASSY

and r/E S e c t , it is C~(B) linear in 5, and Mso the property V~fr/--- [(A o 5)f]r/+ fVor/,

O)

f E C~(B)

is satisfied. (Sec~ denotes the family of all sections o f ~ over B, and rB is the tangent bundle of B). For details we refer to the paper [11]. T h r o u g h o u t the paper manifolds are supposed to be p a r a c o m p a c t , and mappings, vector and tensor fields, functions, etc. to be of class C ~ . B) Some basic notions and facts. Let ~ and ~ be two vector bundles with base spaces B and B I, and let qa : ~ --~ ~l be a bundle map. T h e induced m a p B ~ B ~ determined by qa is d e n o t e d by ~ , and the restriction of ¢2 to the fiber ~r-l(x) is denoted by ~x = qol~r-~(x). We assume each qax to be a linear isomorphism. In this case qo# : Sec ~' ~ Sec ~ is given by [?#(r')](z) = ~-a(r'(~(z))),

r ' E Sec{'.

~p# is additive and satisfies the relation (2)

¢fl#(f'r') = (f' o ~)qa#(v'),

r' e S e c t ' ,

f' E C~(B').

The sections qa#(v') and r' are called V-related. Let (3)

0

,

"

be an exact sequence, where . is the symbol of the pullback. Since I m )~ is Y~ (the vertical subbundle of "rE), therefore ~ : 7r*(~) --. V~, ~ ( ~ ) : = )~(r/) is an isomorphism. Let a : V~ ~ ~, a = pr 2 o ~-~ with pr 2 denoting the projection of 7r*(~) on its second factor, a is a bundle m a p , and its induced m a p is 7r. We fix a horizontal m a p 7~ : 7r*(VB) --+ rE of (3). I m ~ =: H~ is the horizontal subbundle of r E. T h e n an element X of the vector space 9~(E) of the tangent vector fields of E uniquely splits into H X C Sec H~ and V X C E S e c Y , . a# : Sec~ --* SecV~ and (pr2 o #]HE)# = (~r.IHE)# : X(B) --* S e c H ~ are mappings of sections, a#(r/) is the vertical lift of r/ C Sec~ denoted by r/v, and (Tr.IHE)#(X) is the horizontal lift of X e 3~(B) denoted by X h. T h e following facts are well known (cf. [3]): (i) r , X h = X , a o r/v = r/o 7r (ii) x h ( f o 7r) = ( X f o 7r), r f ' ( f o 7r) = 0 (iii) [r/°, ("] = O, H[X h, yh] = [X, y]h,

X, Y E X ( B ) ,

%(CSec~,

Acta Mathema~tica Hungarica 59, 1992

f~C~(B),

(forr) EC~(E).

Y A N O - - L E D G E R C O N N E C T I O N AND I N D U C E D C O N N E C T I O N

407

§2 P u l l b a c k o f t h e m a p p i n g A' 1. In §§2 and 3 let ~ and ~ (resp. ~' and ~') be two vector bundles with a c o m m o n base space B (resp. B~). Let A' : ~ --~ rs, be a strong vector bundle mapping, and ~ : ~ ~ ~', X : ~ ~ ~ two vector bundle mappings with common induced mapping ~ : B ---* B ~. We assume each ~x and X~ (x • B) to be linear isomorphisms. We call a strong vector bundle mapping A : ~ --* rB the pullback of A ~ if A o X#(5 ') and A ~ o 5 ~ are ~ - r e l a t e d vector fields for any 5 ~ E Sec ~ ; i.e. if (4)

Vg' E C ~ ( B ' ) ,

[(A'05')g']o~=[Aox#(5')](g'o~l)

5' ESec~',

or equivalently ( A ' o S ' ) ( ~ ( x ) ) = d ~ [ ( A o X# ( 5') )( x )],

VS' E S e c t ' .

We want to show the existence of such an A, and determine its form. After this we want to investigate the same questions for the pullback of a pseudoconnection (V', A') in ~' w.r.t. ~'. Fig. 1 shows the most relevant ones of the above mappings.

TB

d~

~ TB'

k

B

~ BI Fig. 1

If ~ ( B ) # B', t h e n B' \ ~ ( B ) plays no role in the investigations concerning pullback. So it means no restriction to assume t h a t kO(B) = B I. Surjectivity of the differential d~Px will also be assumed. In order to prove the existence of A we consider the pullbacks ~*(vs,), • *(~) and the strong vector bundle mappings F : r , ~ ~*(rB,),

F ( x , y ) = (x,d~x(y)) • B ×~ TB',

y E ~l(x)

and K :~ -~ ~*(~'),

g ( x , ~ ) = ( x , x ( 5 ) ) • B ×~ E'. Acta l~lathemat~ca Hungarica 59, 1992

408

Z. KOV.~C$, L. TAMASSY F

,s

rB' Fig. 2

Since F is surjective for any Then d~ = pr2oF andx = pr2oK. x E B, there exists at least one strong vector bundle m a p p i n g q~ : ~*(rB,) rB such t h a t F o ¢ = id ([5] Vol. 1, p. 77, L e m m a III). We show t h a t A = ~ o (id × A') o g satisfies (5):

dqYx[A o x#(5')(x)] = =

o(id × A')o

o

=

= p r 2 o F o # o ( i d × A')[x,x(x#(~/))(~(x))]= = pr2 o(id × A')[x,~r'(~(x))] = A ' ( # ' ( ~ ( x ) ) ) ,

VS' • S e c t ' .

This proves the existence of A. 2. T h e local expression of A shows its dependence on A r and gives the grade of its arbitrariness. Since Xx is an isomorphism, we have r a n k ~ = r a n k S ' ( = s), and since is a submersion, we have n = dim B > dim B' = m. Let U and U' = ~ ( U ) be coordinate neighbourhoods in B, resp. B' with local coordinates (z i) (i, j = 1, 2 , . . . , n) and (z ~) (a, 13 = 1, 2 , . . . , m). For appropriately chosen (z) and (z), ~ l u is described by z a o ~ = z ~. Thus dk~o-~(x ) = o°-~(~(x)), and dq)xo- ~(x ) = 0 (~ = m + 1 , . . . , n ) . Let A be a m a p p i n g satisfying (5), and aa-r E Sec~'lu, (a,b = 1 , 2 , . . . ,s) such t h a t they form a basis in any fiber of ~' over U'. T h e n X#(Sla) =: aa have the same property in the fibers of ~ over U. Finally A o 5a = V" z.~i A ai o_~ ox,, where A a are the components of the m a p p i n g A : # - l ( x ) - - . T~B in the frames (o--~,#a). Similarly A ' o a -, ~ = ~ A,~~ o_o ~" According to (5) (6)

(A' o6"a)q~(x) = d ~ ( A oSa ).

Expressing b o t h sides in the coordinate systems described we obtain (7)

A~(x) = A'~(~(x)).

for any A satisfying (5). Conversely, (7) imphes (6) for a = 1 , 2 , . . . , ~ , which is equivalent to (5). This yields however t h a t the components A~, = m + 1 , . . . , n of A(x) can be chosen freely. Our result is expressed by Acta Ma~hematica Hungarica 59, 1992

Y A N O - L E D G E R C O N N E C T I O N AND I N D U C E D C O N N E C T I O N

409

THEOREM 1. a) Given two vector bundles ~ and ~' and a strong vector bundle mappiny A' : ~t __, vs,, as at the beginning of this paragraph, then there exists a strong vector bundle mapping A : ~ ~ vB (the pullback of A') such that A and A' are ql-related. b) The local components A~(x) of this A are determined by (7), and its other components A

(x) are arbit

u. In case of n = m there is no A

(x)

and A is uniquely determined by (7). 1. Section 2 of this § is a local, but constructive construction of all possible A, thus it means a local existence proof of A too. Section 1 was a relatively short and global proof of the existence of A. We remark that Section 2 can be extended to another proof of the global existence of A. Let {U,], ~ E 2" be a local finite open covering of B and {a,], a, E C ~ ( B ) a partition of unity subordinate to {U,]. Let A, : ~l*-*{u,) ~ rBI,~-~(u,) be mappings satifying (4) over U~. Such A, were constructed in Secion 2. We define a,g~ over U, as (a~d,)(5) = at(A,(~)), and as (a~A,)(gr) - 0 on the complement of Ut. Thus atAt is a differentiable mapping taking ~ into rB, and satisfying

[a,A, o X#(bt)l(f'o ~t) = ([a,A'o#']f')o V~' • Sect',

f' • C~(B').

The sum of these yields

Here ~ a, = 1. Hence .4 := ~ ( a , A ~ ) globally satisfies (4), and so it is a pullback of A'.

§3. P u l l b a c k

of the

pseudoconnection

(V',A')

1. Let (V t, A t) be a pseudoconnection in ~' w.r.t. ~t. We call a pseudoconnection ( V , A ) i n ~ w.r.t ~ a pullback of (Vt, X ) 1 if

(8)

Vx#(~,)9~#(~; '): ~ #"~' '" Vb'E Sect, ~;'ESec ('. Lye, r/),

We show that for any pseudoconnection (V', A') there exist puUbacks (V, A), where A must be a pullback of A'. This A is not uniquely determined by i Notions for linear connections more or less analogous to this can be found e.g. in [4] p. 363, or in [5] II, p. 324. Ac~a Ma~hematica Hungarica 59, 1992

410

z. KOVACS, L. TAMA$$Y

(V ~, A ~) except the case dim B = dim B', but ~7 is unique provided we fix a pullback A of X . First we show that A must satisfy (4). We shall denote the pullback X#(5 ') of a 5' ~ Sect' by ~1, similarly ~#(y') = ~h for ~7' ~ Sect', and ~p#f' = f ' o ~ - fl for f ' ~ C°°(B'). Since both (V',A') and (V, A) are pseudoconnetions, in view of (8) we obtain Vx#(~,)~#(f'~' ) = V ~ flO~ = [(A o Crl)fl]~]l + fl V~I ~1 "-~ = ~#(V~,f'~?') = ~#{[(A' o 5')f']y'} + ~ # ( f ' V a , ~ ' ) =

= {[(A' o#')f'] o ~)~1+ f~V~7~,

W' e Sec~',f'

eC°°(B').

Comparison of the first terms in the third and sixth expressions gives (4).

2. Now we construct V and show its uniqueness. In view of (8) the definition of V for 5~ and ~1 can only be (9)

V~1~71 -- (p#(V~,~?'),

V~I e Sect,

7/1 E Sec¢.

We show that the value of V over sections "deferring" from pullback sections must be 0. Thus we have only one possibility for the definition of V. Finally we show that the V defined in this single possible way coupled with an A satisfying (4) is a pseudoconnection. We define V at an arbitary point x0 E B for any 5 E Sec ~ and 7? E Sec ~. Let B1 C B be a (possibly small) submanifold of B through x0 such that • : B] --* ¢(B1) - B~ C B' is a bijection and A o 5(Xo) e TxoB1. Clearly B1 is not uniquely determined by these conditions. We define on B~ y' := ~(y ],,), 5' := X(51,~) and on ~ - l [ B i ] Yl = ~2#(77'), 7]2 = ~ - T]I; al = X#(al), a2 = a - al. Thus any 5, or ~7splits w.r.t, the chosen B1 as

(10)

5 : ~1 + ~2,

~ : ~1 + ~2.

We have the relations

(11)

a) al-

=

b) , i , , =

c)

=

= O.

Assume (V, A) to be a pseudoconnection satisfying (8). Then, because of the additivity of V

(12) on B1. For any linear connection V L in ~ V#~?- VLoar] = T(5, ~) is a function-bilinear mapping Sec ~ x Sec ( ~ Sec ~, i.e. T is a homomorphism: T E Hom(Sec~, Sect; Sect). Hence we obtain on B1 a representation of V in the form + Ac~a Mathematica Hungarica 59, 199P

YANO--LEDGER CONNECTION AND INDUCED CONNECTION

and

411

L

V ~ 711 = VAo~ 711 + T(~I, 711). However b(Zo) = bl(zo), from this A o b(Zo) = A o bl(XO), f u r t h e r m o r e =

M o r e o v e r o n B1

=

we =sumed

A o

T~oB~. F r o m these we obtain V~/[~ 0 = V~/1]~0. Hence the sum of the last three terms in (12) is zero at xo. Moreover this s u m m u s t vanish for given ~l, 711 and for arbitrary b2, ~2 satisfying (11.c). Thus these three terms m u s t vanish also separately at xo. Hence the definition of a V satisfying (8) for arbitrary 5, and 7] can only be (13)

V87] = V~711 ---=~ # ( V ~ , 7 ] ' ) .

We alo obtain t h a t (14)

V~2711 = V#~ 7]2 = V#2712 = 0

which are consequences of (13). In this definition we m a d e use of the splitting (10), which depends on the choice of B1. Using another B1 (satisfying the same conditions) we obtain another splitting 5 = 51 + 52~ 7] = 711 nu 7]2 and then (15)

V~7] = Val T/1 = ~#(V;,7]') = ~ # [vI"'LA,o~,7]'+ T(~', 7]')]

resp. (16)

• ---~ V~7] "-- V._ 711 tr1

( V t . ; 1) --= ~1

V'

1o

T(

;,,')] 7]!

at x0. But b 1 -- ~rI and ~h = ~}1 at z0, resp. in the direction of A o b(Zo). Consequently the same hold true also for a', - 5' and 7]', 7]1 at ~ ( z o ) = x~, resp. in the direction of A o ~'(x~). Therefore the right h a n d sides of (15) and (16) equal. Hence V~lT]1 = V • 711 at x0, and thus the definition of V 81

does not d e p e n d from the choice of B1. 3. We have still to show t h a t the (V, A) just constructed is a pseudoconnection. V is additive b o t h in 5 and 7], for V ~ is so in b~ and ~/'. Consider now for an xo, ~r and for a B1 corresponding to these zo, ~ a correspondence x ~ zl = 9 - 1 [ ~ ( z ) ] n B1 (z E B). Define for an f E C°~(B) the functions f l ( x ) = f ( x l ) , f2(x) = f ( x ) - f l ( z ) , and f ' E C°~(B ') as f ' = f l o qJ. T h e n f 2 [ B 1 - - 0 and f a l = f l ~ l + f2~rl. Here V1 ------f1(71 is a pullback field, and f2al is a P2, for (/2hi)Is1 = 0. In view of (13) and (14) VIS- ~ ---- ~7f~.1 ~ 1 : Vp~..'r]l -[- Vp.z 711 ---- Cp#(~f,5.,r]" ) = ~0# (fIVs.,rIs ,.) ~_- f1~75.1 7]1 Acta Mathematica Hungarica 59, 199~

412

z. KOVACS, L. TAMa{SSY

at Xo. We m a y we have at x0 V~fr] = V & f ~ l of (14) Valf2r]l

add f 2 V a l r h to the last expression, since f2(x0) = 0. Thus V l ~ l r h -- fV~lr]l i.e. V is C~(B) linear in 8. Finally = V ~ f V h + Valf2rh. Here f2r h is again a P2, and in view = 0 at Xo. Thus in view of (4) and (13) valf~l

= v~lfl~l

= ~

#

!

! I

(v~,f ~ ) =

= ~o# {[(A' o 5 ' ) f ' ] r]'+ f ' V ' , r / ' } = [(A o 51)fl] r]l + flVslr]l. We can add again at xo f 2 V ~ l and [(A o 51)f2] 71 to the right h a n d side, since A o 51(Xo) e T, oB1, f2]B~ = 0, and hence [(A o 51)f2]]x0 = 0. Thus V ~ f r h = [(Ao 51)f]r h + fV#~r h at Xo, i.e. V satisfies (1). These show t h a t (~7, A) is indeed a pseudoconnection. We have the following THEOREM 2. a) I f ( V , A) is the pullback of the pseudoconnection (V',A'), then A must be a pullback of A'. b) If (V', A') is a pseudoconnection, and A is a pullback of A', then there exists a unique pullback pseudoconnection (V, A), whose V is determined by (13). If we want to emphasize t h a t (V, A) is pullback of a (V', A') t h e n we write V '# in place of V.

4. V : X(B) x X(B) --+ X(B), (X, V) ~ V x Y = 0 is clearly no linear connection over B, for it does not satisfy the fourth Koszul axiom V x f Y = = ( Z f ) Y + f V x Y , f E C~(B). But for a pseudoconnection (V, d ) , where A : ~ --+ 0 E rB (i.e. A is trivial) V~r] = vLoa~ with a linear connection V L on B (in this case we say t h a t V is associated to V L [6]), we have Va~ = 0, Vb, r]. We want to s t u d y a similar case. PRo osmoN 1. given pseudoconnections (V, A) such that

(17)

Vx#(~,)~,o#(r/' ) = O,

X and A :

Vb' E S e c t ' ,

-+

there e ist

r]' E Sect'.

In this case Aox#(5')(x)CKerd~ltx,

(18)

PROOF. Define in ~

a VL

Vb'eSec~',

and a h o m o m o r p h i s m T ( & ~?) such t h a t

L

VAo8xr]l = T(51, r]l) = 0. Then + Acta Mathematica Hungarica 59, 199~

xEB.

YANO-LEDGER

C O N N E C T I O N AND I N D U C E D C O N N E C T I O N

413

satisfies (17), and (V, A) is a pseudoconnection. Definition of the linear connection V L and of the homomorphism T for any other 5 and T/may be arbitrary. Thus (V, A) need not be an associated pseudoconnection. For a pseudoconnection (V, A) we have

v~/~nl = [(A o ~1)f~]~1 + I~Va~n~, Vf1,¢1. From this and (17) we get (A o 5~)fl = 0, Vf~. According to the derivation of f l from an f E C ~ ( B ) f l is constant on every ~ - l [ z ' ] , otherwise it can be arbitrary. So X E TxB is tangent to ~-~[~(z)] iff X f x = 0, Vf~. Hence A o 5i must be such an X , i.e. A o 5i(x) E Ker d ~ . Q.E.D. (18) is obviously a weaker condition than A : ~ ~ 0. Hence, in case of Ker d$:~ # 0, (17) may be satisfied even if A is not trivial.

§4. I n d u c e d v e c t o r b u n d l e F i n s l e r c o n n e c t i o n s 1. M. M a t s u m o t o [7] (see also P. Dombrowski [2]) considers a Finsler connection on B as a pair (7~, V) of a horizontal map 7-/to T T B and of a linear connection V in the vertical subbundle VrB of the tangent bundle VB. He induces a Finsler connection (7~, V) from a linear connection V of B (from a single data), and calls it the V-linear Finsler connection. He also extends a Finsler connection (7-/, V) to a linear connection V' on T M (in T T M ) . In this paragraph we consider vector bundle Finsler connection in the sense of R. Miron [8], and by making use of the just established Theorem 2 we derive an induced vector bundle Finsler connection from three data (sources). In a special case this gives the extended Finsler connection V' of Matsumoto. Let ~ = (E, ~r,B, V r) be a vector bundle, 7"l : ~ * ( T B ) .---+ TE a horizontal map, I I ( = ( l i E , 1rE, E, V") the horizontal subbundle, V ( = (V E, 7rE, E, V") the vertical subbundle of rE; h resp. v the horizontal, resp. vertical lifts, as in §1. Consider four pseudoconnections

(V 1, H) (v2,ii) (19)

v l : X(E) × XH(E) -~ XH(E) V 2 : :E(E) × X v ( E ) --+ X v ( E )

(v3,v) v3: X(E) × Xv(E) ~ Xv(E) (v4,v) V4: X(E) × X . ( E ) ~ XH(E) N.v(E) = See V~,

:~H(E) = Sec H~.

As well known, a nnear connection V : X(E) × X(E) --+ X(E) is a

Finder connection in the sense of Miron [8] w.r.t. 7-/if V x H Y is always a horizontal and V x V Y a vertical vector. It is easy to see that any such Finsler connection can be represented in the form

(2o)

V x r = v~.iiv + v ~ x v r + v ~ v r + v~iiv, Acta Mathema~ica Hungarica 59, 199~

414

z. KOVA.CS, L. TAMASSY

where ~ 7 1 , . . . , ~74 are pseudoconnections of type (19), and conversely, any sum as on the right hand side of (20) is a Finsler connection w.r.t 7-/in the sense of Miron. This result formulated in another way can be found in [11]. TItEOttEM 3. Consider a tangent bundle rB with a linear connection

vr:

X(B) x X(B) -+ X(B),

a vector bundle ~ = ( E , r , B, V r) with a linear connection

V L : :~(B) X See ~ --+ See on it, and a horizontal map

~/: 7r*(rB) -+ rE. Then there exists a unique Finsler connection V : T ( E ) x Y.(E) --+ T ( E ) w.r.t TI such that

(21)

a) b)

Vx"r/" = (V~fr/).

c), d)

V a r y h = Va*rl v = 0

V x h Y h : (VTxy)tr X,Y

e X(B)

a, r/E See ~.

P R o o f . Let us specialize the objects ~, ~, {', ~', % X, V', A' and A playing a role in the considerations of §3 in the four case a) to d) as

~

a) b) c)

H# V{

H,~ H#

V{

d)

H~

~)

b) c) d)

~( 7r. rr. ~' a

,"

~'

v

rB rB

V{

{

~

~

v

V~

r~

~

~,

h

X# h h

W VT VL

v v

-

~r. a

~#

rB ~

AI A id H id H -

V V

h v

V V 1# V 2# Ca ~,4

(the roles of B, B' and • are assumed in each case by E, B and rr). Then in cases a), b) according to Theorem 2, and in cases c), d) according to Proposition 1, there exist unique pullback pseudoconnections (V 1#, H), (V 2#, H) and pseudoconnections (~73, V) and (Q4, V) with the following properties:

(22)

a)

V 1• : ~'H(E) X ~.H(E) =-+ :~.H(E)

b)

V 2# : X H ( E ) × X v ( E ) --+ Xv(E)

c)

~7a

: :~v(E) x X v ( E ) --+ :~v(E)

d) ~4 :Xv(E) x EH(E) --* Y.H(E) Acta M a t h e m a t i c a Hungarica 59, 1992

Xh--

--

V2# . + ,_~ = ( v ~ , ) " -3

v

-4 h V,,~Y =0.

Y A N O - - L E D G E R C O N N E C T I O N AND I N D U C E D C O N N E C T I O N

415

Hence

v b H z = v' V ~ V Z = £7~uVZ ,

Hz,

vbvz = %

V b H Z = fT~uHZ,

vz, U,Z C X(E)

together with the mapping H, or V are four pseudoconnections as those of (19). Thus the sum (20) formed from these last pseudoconnections yields a Finsler connection (23)

V u Z = VaHuHZ + V2HuVZ + V ~ u V Z + V~.uHZ ,

U,Z C X(E).

Making use of the listed properties of the pseudoconnections (22) one can easily check that the constructed Finsler connection (23) satisfies the required properties (21) of the theorem. We show still the uniqueness of this Finsler connection V. Suppose namely that V' is another Finsler connection with properties (21). Then the difference tensor D(U, Z) = V u Z - V~rZ vanishes for any vertical and horizontal lifts: D ( X h , Y h) = D(Xh, a ~) = D ( y ' , Y h) = D ( a ~ , ~ f ) = 0. Hence D ___-0, and V x Y = V ~ Y . Q.E.D. The problem of Theorem 3 is dealt with by different method also in [11]. The Finsler connection V defined by (23) is deduced from three independent data V T, V L, T( as given at the beginning of our theorem, and thus this V is a Finsler connection induced from three sources. It can be denoted by V -- (~TT, v L , ~-~). Assuming that 7/is the horizontal mapping 7/L o f V L, or equivalently that 7~ satisfies the homogeneity condition (see [9] p.311) and hence 7-I = 7~L ¢=~ V L, we obtain a Finsler connection (vT~.~'~L) induced by two sources. If ~ = rB, V T = V L and 7-I = ~.~L ~ ~7L ' then (V T, V L, T/) is the extended Finsler connection V t of a V-linear Finsler connection (7"/,V.) of Matsumoto. Thus (V T, V L, 7"/) can be conidered a generalization of this V' to a vector bundle Finsler connection. A simple local calculation gives the PROPOSITION 2. In an induced Finsler connection V = (V T, 7-/) Vxh~ ~ = [xh,~].

PROOF. It is easy to see that (V, h), where

:~ : XH(E) x Xv(E) -+ Xv(E),

(xh,,7 v) ~ Y[Xh,,7"],

is a pseudoconnection. We want to show that Vx.~ =

(v}~) ~, Acta Mathematica Hungarica 59, 199~

416

z. KOV~,CS, L. TAMk.SSY

where V L is the linear connection determined by ~-~L. Let (xi, yU), i = 1 , 2 , . . . , n ; # , v = 1 , 2 , . . . , r = rank~ be a local coordinate system on 7r-l(U), U C B, and eu E Sec(Iv a base in each r - l ( x ) , x C U: Let N~(x, y) be the local components of 7-/L, i.e.

and r ~ the local components of ~i~TL. Then

r5 = oy.. For X e X(B)

x l . = x' o o. "

Xhl"-*(~) = ( x ~ o ,r)

( ~ o _ N."A' av.] '

O and for r/E S e c t , r/Iu = r/Ueu, r/" = (r/U o 7r )o-b-~,

IX h, ¢1 =

[

( x ' o ~)

( 0 _N.o ~

' o y . ] ' ( ~f o

0]

Computing this and reforming the result we obtain

f

[xh,¢]=(x , o ~ ) 0 ( ~ ] ~ )

- -0 + (x~ o ~ ) ( ¢ o ~)r~; 0

OyU

= {x'°U"~ \ ~ u + x ' T' r~i ~"e "

OyU

It/ = (v~.r,

where ~7L is the linear connection uniquely determined by ~.~L. The last expression is vertical vector. Thus

(z4)

Vx.,r

= v [ x h , , T q = [ x h , ¢ ] = (v],7) ".

The induced Finsler connection V = (~ 7T, 7-~L) of our Proposition is clearly the same as V = (V T, V L, 7"/L) of Theorem 3, for V T and ~.~L are the same in both connections and V L is determined by 7~L. Hence we can apply the statement (21,b) of Theorem 3 which together with (24) yields the statement of the Proposition. Q.E.D. Acta Mathematica Hungarica 59, 1992

YANO-LEDGER

CONNECTION

AND INDUCED

CONNECTION

417

§5. Yano-Ledger connection on vector bundles Starting with a linear connection on a manifold M, K. Yano and A. J. Ledger [13] constructed a unique torsion-free linear connection on T M satisfying certain simple conditions. We perform the same replacing T M by the total space E of a vector bundle ~, and replacing M by the base space B of ~. Thus our result gives back Yano and Ledger's theorem, provided = rB. Moreover, the resulting connection turns out to be an induced vector bundle Finsler connection in case if it is torsion-free, and its symmetrisation otherwise. Our proof is completely different from that of Yano and Ledger's theorem, not only for the difference between the tangent bundle used by Yano and Ledger, and the vector bundle we have considered, but rather because we arrive to our result on a quite different way. THEOREM 4. Let V T : ~ . ( B ) X ~ ( B ) ---* X ( B ) be a linear connection o n the base space B of a vector bundle ~ = (E, 7r, B, Vr), and let rt " r*(rB) ~ r~

be a fixed horizontal map satisfying the homogeneity condition [9]. Then there is exactly one torsion-free linear connection o

v : x(E) x X(E) ~ X(E)

satisfying the conditions o

(25)

v ~ o , " = 0,

(26)

v~rh

o

o

(27)

Vxhyh=

= o,

( V~Y-~Wor(X,Y)

+

v [ x h , y h]

T

Where X , Y E X(B); a, rI E Sec ~; and Tor denotes the torsion of V T. PROOF. Let V = (V T, 7-l) be the Finsler connection induced by V T and 7"( (see §4), and let o VvZ = VvZ-

(28)

1 v ~Tor(V,Z),

V , Z E Y.(E)

Then o

VvZ =

(VvZ+ V z V

+ [V,Z]),

o

a . d V satisnes (25)-(27). Indeed, in view of (21,d) ~r~ r/" = ~1( V ~ o , ,~ + V~oa ~ + [a~,,~]) = ~[a 1 ,, ,,~]. Acta Matkematica Hungarlca 59, 199~

418

z. KOVACS, L. TAMASSY

Vanishing of this bracket is a known fact (see (iii) at the end of §1.) Hence o

V ~ r f = 0. Again, in view of (21,c) and Proposition 2 1 "V Vo , "Yh= ~( Y~ a v +[a~,Yh])=

1

([Yh, ao]+[av,Yh])=O.

Finally in view of (21,a)

Vo x ~ Y

h

= ~1 {(VffcY) h + ( v T x ) h + [ x h , y h ] } .

Splitting the last term in a vertical and a horizontal part, and taking into account that H[X h, yh] = [X, y]h, since X h and yh are r-related to X and Y, we obtain

o

VxhY h =

{1 VTxY

= (vTy

-Jr ~ T y X q- ~[X,Y] 1 r

+

y[xh,y

h] =

h+

These prove the existence part of our theorem. A short and straightforward o

calculation using (28) shows that V is torsion-free. o

In order to prove the unicity we note that from Tor(Y h, a v) = 0 and from (26) we obtain

(29)

=

+ [rh,

o] = [yh,o.].

Now suppose that V is another torsion-free linear connection satisfying (25)(27). Then it satisfies (29) too. Hence the difference tensor o

*

D(U, V) = V v Z - V u Z vanishes for any horizontal and vertical lifts:

D(xh, y h) = D(Xh,a ~) = D(tr',Y h) = D ( a ' , r f ) = 0, and thus D = 0. Q.E.D. O

O

In case ~ = vB, V T = V L, V is the Yano-Ledger connection. Thus V can be called a Yano-Ledger connection on vector bundles. Furthermore, from Proposition 2 and (24) we obtain o

COROLLARY 1. Vxhr f = (V~r/) ~.

Since a horizontal map 7-I satisfying the homogeneity condition determines a unique linear connection V L, and conversely, so we may denote it by 7-/L, and we obtain the following Acta Mcthematica Hungarica 59, 199~

YANO-LEDGER CONNECTION AND INDUCED CONNECTION

If the induced Finsler connection

THEOREM 5. V

419

V = (~7 T, V L) has a

V

torsion Tot, then V - ~Tor is the Yano-Ledger connection induced by the same V T and V L (or equivalently 7-lL). If the induced Finsler connection V = (V T, V L) is tor.sion-free, then V = (V T, V L) coincides with the Yahoo Ledger c o n n e c t i o n V ( V T, ~.~L). This means a generalization of Matsumoto's extended connection V ~ (i.e. of the V-linear connection (7/, V) for TM; [7]) to vector bundle Finsler connection and to vector bundle Yano-Ledger connection.

§6. Metrical Y a n o - L e d g e r c o n n e c t i o n o n v e c t o r b u n d l e s Let g be a Riemannian metric in vB (i.e. on B), ~ a Riemannian metric in ~ = (E, % B, Vr), and 7~ : r*(rB) --* rE a horizontal map. Then

Cp( Up, Zp) =

g.(p)(~.U~, ~ . Z ~ ) +

~(~)( ~(VUp), ~(V Z~)),

U,Z E •(E), p E E is a Riemannian metric in rE (i.e. on E; [4]). Clearly

(30)

G(Xh, Y h) = g ( X , Y ) oTr C(~,~) = ~(~,,) o ~ G(Xh, a ~) = O, X , Y E :~(B); a, y E Sec~.

Let V T be the Levi-Civita connection on (B,g), V i a metrical linear connection of (~, ~), and V the Levi-Civita connection on (E, G). Let us derive o

from V T and V L the Yano-Ledger connection V in rE. One can put the question whether or not this Yano-Ledger connection is the Levi-Civita connection in rE. The answer depends on the properties of V L and ~. o

THEOREM 6. The Yano-Ledger connection V (induced by V T and V L) in rE is the Levi-Civita connection in rE iff the Riemann-Christoffel curvature tensor R(a,~?,X,Y) := O ( k ( X , Y ) q , a ) , a, rl E See(; X , Y E X(B) of V L vanishes (R is the curvature map of v L ) . 0

PROOF. We shall first compute V G and then our theorem easily follows. Every vector field U, Z, W E 3C(E) can be written as a finite C°°(E)-linear combination of vertical and horizontal lifts. Therefore it is sufficient to comO pute (VG)(U, Z, W) for vertical and horizontal lifts. Using the explantation

(VG)(U,z ,

w ) - V ~ G ( Z , W ) - G ( V v Z , W ) - G(Z, V v W ) , Acta Mathema~ica Hungarica 59, 1992

420

z. KOVACS, L. TAM/~SSY o

properties (30) of G, and properties (25)-(27), (29) of V we obtain o

(va)(~,

or, 7~) = ~ a ( ~ ,

¢)=

~(~(~,7)o

~ ) = 0,

o

( v a ) ( v r v , x h , y h) = crVG(Xh, y h) -- a ° ( g ( X , Y ) o ~r) = O, o

( v c ) ( ~ ~, 7 ~ , x h) = o. Making use also of Corollary 1 we get

(VC)(xh,~,¢) :o o = x h e ( c r v, 7 v) -- G(Vxhcr v, 7 v) -- e ( o .v, V x h 7 v) = = xh(o(,~",T")O~)- a((V~o)',

7 ~) - a ( ~ " , (VLx~) ~) =

= Xh(~('~, 7) o ~) - ~(VLx~, 7) o ~ - ~(~, VLxT) o ~ : = x h ( ~ ( ~ , 7) ° ~) - (X~(~, 7)) o ~ = 0,

( V G ) ( x h , y h , z h) = x h G ( y h ,

z h) - G ( V x h y h , z

h) - G ( y h, ~Tx h Z h ) =

= X h ( g ( X , Y ) o r) - g(VTx Y, Z) o 7r - g(Y, V ~ Z ) o r = = X h ( g ( X , Y ) o 7r) -- ( X g ( Y ,Z)) o 7r = 0 and

( ~ c ) ( x h, ~ , r h) = - c ( ~ xh~ ~, Yh) - c(~ ~, ~ x.Yh) = "- - G

~rv, W[X h,Yh]

= -~O(cr , -o~ o ViX h,Yh]).

J. Szilasi [10] has shown that (x o V [ X h, yh] o p = R(X, Y)(p), p E Sec ~. Hence

(va)(xh, :--2~(Or, O~0 v[xh,yh]o,

avyh) o p =

p)---- 2 [ l ( o ' , R ( X , Y ) ( p ) ) = R ( o ' , p , X , Y ) .

o

Thus R = 0 ~ V G = 0, i.e. V is the Levi-Civita connection for (E, G). Q.E.D. As well known, the Nijenhuis torsion of 7-[ is

N~(X,Y) Hence

= [HX, HY]-t- H [ X , Y ] -

H [ H X , Y ] - H[X, HY].

NT.l(X h, y h ) _ IX h, yh] _ H [ X h, yh] = y [ x h, yh].

Thus we also get the COROLLARY 2. The Yano-Ledger connection in rE is the Levi-Civi-

ta connection on ( E , G ) iff the Nijenhuis torsion of 7-l vanishes, i.e.

V[X h, yh] = O. Acta Mathematica Hungarica 59, 1992

iff

YANO-LEDGER CONNECTION AND INDUCED CONNECTION

421

References [1] J. Dieudonn~, Treatise on Analysis, Volume 3, Academic Press (New York, 1972). [2] P. Dombrowski, # 3469. MR.37. [3] P. Dombrowski, On the geometry of the tangent bundle, J. Reine Angew. Math., 210 (1962), 73-88. [4] T. V. Due, Sur la geometrie differentielle des fibres vectoriels, Kodai Math. Sere. Rep., 2 6 (1975), 349-408. [5] W. Greub, S. Halperin and R. Vanstone, Connections, Curvature and Cohomology, Volume I, II, Academic Press (New York, 1972, 1973). [6] Z. Kovacs, Some properties of associated pseudoeonnections, Per. Math. Hung. , to appear. [7] M. Matsumoto, Connections, metrics and almost complex structures of tangent bundies, J. Math. Kyoto Univ., 5 (1965-1966), 251-278. [8] R. Miron, Vector bundles Finsler geometry, in Proe. Sere. on Finsler Spaces (Brasov, 1983), pp. 147-188. [9] J. SzilRSi, Horizontal maps with homogeneity conditions, Rend. Palermo, Suppl., 3 (1984), 307-320. [10] J. Szilasi, On the curvature and integrability of horizontal maps, Acta Math. Hung., 46 (1985), 183-188. [11] J. Szilasi and Z. KovKcs, Pscudoconnections and Finsler type connections, in Coll. Math. Soc. Y. Bolyai, 46 (1984), pp. 1165-1184. [12] Y. Wong, Linear connections and quasi connections on a differentiable manifold, Tohoku Math. Y., 14 (1962), 48-63. [13] K. Yano and A. J. Ledger, Linear connections on tangent bundles~ J. London Math. Soc., 39 (1964), 495-500. (Received February 2, 1990)

KOSSUTHLAJOS UNIVERSITY DEPARTMENTOF MATHEMATICS H-4010 DEBRECEN~P.O.BOX 12

Acta Ma~hematica Huu qarica 59, 1992