Almost Complex Structures on Tensor Bundles

. J . DIFFERENTIAL GEOMETRY 1 (1%7) 355-368

ALMOST COMPLEX STRUCTURES ON TENSOR BUNDLES A. J. LEDGER & K. YANO

1.

Introduction

It is well known that the tangent bundle of a Cw manifold M admits an almost complex structure if M admits an affine connection [ I ] , [ 5 ] or an almost complex structure [7], [8]. The main purpose of this paper is to investigate a similar problem for tensor bundles T : M . We prove that if a Riemannian manifold M admits an almost complex structure then so does T:M provided r s is odd, If r $- s is even a further condition is required on M . The proofs depend on some generalizations of the notions of lifting vector fields and derivations on M , which were defined previously only for tangent bundles and cotangent bundles 141, [7], [8], [9], [lo].

+

2. Notations and definitions

M is a C * paracompact manifold of h i t e dimension 1 1 . F ( M ) is the ring of real-valued C” functions on M . For r + s > 0. T:M is the bundle over M of tensors of type ( r , s), contravariant of order r and covariant of order s. 7r is the projection of T;M onto M . We write TbM = T ’ M , T:M = T , M . ,F,;(M) is the module over F ( M ) of C - tensor fields of type ( r , s). We writc ./;,(M) = , T ’ ( M ) ,,=T;(M)= T 8 ( M ) , and ,F!(M) = F ( M ) . X ( M ) is the direct sum C F ; ( M ) . T , is the value at y E M of a r. v tensor field T on M , and ,Ti@) is the vector space of tensors of type ( r , s) at p . Let S E ,=T;(p) and T E . F ; ( p ) . Then the real number S ( T ) = T ( S ) is defined, in the usual way, by contraction. It follows that if S E ,Fs(M) then S is a differentiable function on T I M . A map D : Y ( M ) + .F(M) is a derivation on M if (a) D is linear with respect to constant coefficients, (b) for all r , s , D.P;(M) c 7 ; ( M ) , (c) for all C” tensor fields T , and T 2on M , D ( T , (3 TJ = ( D T , ) 0T 2 -I- T , (3 DT,, Communicated August 15, 1967.

313

356

A. J. LEDGER & K. YANO

(d) D commutes with contraction, A derivation is determined by its action on F ( M ) and F1(M). In particular, . Y i ( M ) may be identified with the set of derivations which map F ( M ) to zero. The set of derivations on M forms a module 9 M over F ( M ) . (vii) The notation for covariant derivatives and curvature tensors is that of [ 2 ] , The linear connections considered on M are assumed to have zero torsion. 3.

Vector fields on T,'M

In this section we show how vector fields on T:M can be induccd from vector fields, tensor fields of type ( r , s). and derivations on M . We first prove a lemma which, together with its corollary, will be of usc later. Lemma 1. Let p E M and S E n - ' ( p ) . If W is LI vertical vector nt S (i.e. tangential to x - ' ( p ) ai S) and W ( n )= 0 for all N E 2 - ; ( p ) then W = 0 . Proof. The vector space . Y , ; ( p ) is dual to F;(p)and hence N contains a system of coordinates on n - ' ( p ) , The result follows irnmcdiately. Corollary 1. Let W E S ' ( T : M ) . If W ( N ) =for ~ all N E f ; ( M ) then W = 0 . Proof. The assumption on W implies that for 5, E 3 - - l ( M ) and f E F ( M ) , 0=

I W(rlf203 ) = W ( ( f 7r)df013) = W ( f . 7r)tlf 0,3 . 2

-~

L

Hcncc (IxW = 0, and so W is a vertical vcctor field. Thus W = 0 by Leniina 1. thc values of W on the zcro section of .F;M being zero by continuity. Proposition 1. Let T E 3-:(M). Then there is u irniqrre C,' vector field T" on TIM siich t h t for N E J-;(M), T'(tu) = n ( T ) n

(1)

L

,

Proof. For p E M , n-I(p) is a vector spacc and so T,]determines B unique vcrtical vector field TY, on n-I(p) such that for N E c ~ - ; ( p )T;:(n) , = (y(T1,).The cross section T on T:M then determines a C' vcrtical vector field which satisfies ( 1 ) . T" will be called thc verticul lift of T , Corollary 2. Let S E r ' ( p ) , anti let T i be the valire of T' at S. Tlieri the map T,, T ; is a linear isomorphism of n - ' ( p ) (n-l(p)).,, where ( n - ' ( ~ ) ) . ~ is the tangent space to the fibre n - ' ( p ) at S . Proposition 2. Let D be a derivation on M . Tlieri there is a irniqrre Liector field D oti T;M such that for N E Y ; ( M ) --f

(2)

--f

Drr = Dru

314

,

3 57

ALMOST COMPLEX STRUCTURES

-

Proof. Let {P}(i = 1, 2, . ., n) be a coordinate system on a neighbourhood U of p E M , and {a8}(0 = 1, 2, . . . , nr+s)a basis for F ; ( U ) . Then {xz 0 n,d}is a coordinate system on r - ' ( U ) . Define D on n-'(U) by (3)

D(X1

c

n) = (DX') 0

,

71

D(wR)= D(oe) .

(4)

Thus a C" vector field D is defined on x L 1 ( U ) . Moreover, for (Y E S f ( U ) we have Dlru = Dlru. Hence, using Corollary 1, it follows that D is defined over YM as the unique solution of (2). Corollary 3. If f E F ( M ) then D(f c n) = (of) IC. Corollary 4. D is a vertical vector field if and only if D E F ; ( M ) . Corollary 5. If D,, D2 are derivations on M , and f l , f 2 E F ( M ) , then f,D, + fzD, is a derivation on M , and Q

__ _ _

f a

+ f?D, =

K)DI

(fl

t (f, n>D2 .

Thus if F ( M ) is identified with F ( M ) n IC = {f o IT : f E F ( M ) } flier1 D linear map of .QM -+ P ( T ; M ) . then for S E Ti([)), Corollary 6. If p E M and A E

A,

( 5 )

-t

D is

(I

= -(&)I, ,

where the sirfix S tletiote~evaliration at S. Proof. Let lru E . Y ; ( p ) . Then

A,(cu>= ( A t r ) ( J ) =

-(AS)L((Y) .

'I he result follows from Lemma 1 . denote Lie derivation with respect Corollary 7. Let X E T 1 ( M )arid FA, to X . Then 2,i.r a vector field on T ; M . In conformity with the notation of [4], [8], [9], [lo], we call 2, the complete Lft of X and write 9,= A''. Remark 1 . If f E F ( M ) then Y , k

where X (6)

0df

=

f 9 1

-

XOcif,

is regarded as a derivation on M . Thus

(fx).= (f

L

n)XC -

-~

x 0clf

Now if T:M is the tangent bundle T ' M thcn for

LY

E

x 0df(n) = - a ( X ) d j . I-ience by Proposition 1 ,

315

*

F1(M),

358

A.

J. LEDGER 8: K. YANO

____

X Q df = - d f X v , where X" is the vertical lift of X to TIM. We then have

(7)

(fX)C = fXC

+ dfX" .

Equation (7) was used extensively in [8] but does not appear to extend to tensor bundles of high order. Equation (6) is perhaps a useful generalization. Lemma 2. Let p E M and A E FT:(p). Suppose there exist non-negative integers a and b , not both zero, siiclz that A F ; ( p ) = 0. Then A = kZ where k is some real number. If a $. h then A = 0 . Proof. We prove the lemma for the case a > 0. The proof for a = 0 and b > 0 is essentially the same with covariance and contravariance exchanged. Let S E F ; - ' ( p )be non-zero, and let X E F-'(p). Then AS @ X

+ S@AX

=0

.

Choose (0 E 2;;, ( / I ) such that w ( S ) # 0. Then ( A - kZ)X = 0, where k = - w(AS),'o(S). I t follows immediately that A = kZ. Then for T E ~ - ; ( y ) 0 = A T = I\(N

b)T.

-

Hence, if N # b then A 0 and A = 0. Remark 2. A = XI for some X is a necessary and suflicient condition for A f ; ( p ) = 0, CI # 0 . Corollary 8. Let 1) E 9 M antl si~pposcthew e x i s t non-negntii3eintegers a antl b , tzot borli ;ern, ~irclztllrit D , I ; ( M ) = 0. Then D = f!, where f E F ( M ) . If a + b then D = 0. Proof. Let Ii E F ( M ) and T E J ; ( M ) . Then

(Dh)7= 0

,

It follows immediately that D F ( M ) = 0 antl hencc D E J : ( M ) . Then by Lemma 2, D = fZ for some f E F ( M ) , and if CI # b , then f is zero by Lemma 2. This completcs the proof. Remark 3. D = fZ for some f E F ( M ) is a necessary and suflicient condition for D 5 ; ( M ) = 0, a # 0 . Corollary 9. The map D -> D of (irM --* S ' ( T ; M ) is a nioiionioryliisrn when r f s antl has kernel { f I : f E F ( M ) } wlieri r = s. Proof. This follows from Corollaries 1, 5 and 8. Corollary 10. I f r # s then TvM atlnzits N vertical vector field which vanishes only on the zero section of T ; M , Proof. The vector ficld f has the required properties. Corollary 11. Let p E M , A E f.jT:(p) and T E F ; ( P ) , r # s. Theti A = T v implies A = 0 and T = 0 .

316

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ALMOST COMPLEX STRUCTURES

Proof. Suppose 2 = T". Then by Corollaries 2 and 6, AS = -T for all S E F ; ( p ) . Since A is linear it follows that T = 0 and A T ; @ ) = 0. Hence A = 0 by Lemma 2. Suppose now that r is a linear connection (with zero torsion) on M , and let X E P ( M ) . Then FX E Y ; ( M ) , and hence, by Corollary 4 , is a C" vertical vector field on G M . Another C" vector field on T;M is determined by the derivation V , . In conformity with [4] we write r, = X h , and call X " the horizontal lift of X . If f E F ( M ) then using Corollary 3,

r,

X / ' ( f ' n ) = r,(f n ) = ( C , f ) n = (Xf) n . Q

Hence dnX" =

(8)

x.

The horizontal lift clearly satisfies

(fxf

gy)" = ( f

G

n)xh+ (g

0

n)Yh ,

for f , g E F ( M ) and X , Y E F ( M ) . Thus the horizontal lift is a linear map of F ( M ) --t P ( P s M ) if, as before, F ( M ) and F ( M ) o n are identified. Since = 0 if and only if X = 0, the horizontal lift is a monomorphism, and so determines a horizontal subspace H , of dimension n( = dim M ) at each point S E T i M . Then C" distribution H on T i M so obtained is usually calIed the horizontal distribution determined by the connection r . If S E T:M then the tangent space ( P s M ) . yis the direct sum V,s H s , where V Sis the subspace of vertical vectors at S . Thus, if W E (T:M),ythen

e,

+

w = h(W) + V ( W ) , where h and 'u are the projections onto the horizontal and vertical subspaces at S. Clearly XIt = Iz(X[l) and Tv= v ( T u )for any vector X and tensor T of type ( r , s) at n(S). 4.

Lie brackets

We now determine, for later use, the Lie brackets of some particular types of vector fields on T;M. These results generalize some of those already obtained for tangent bundles and cotangent bundles [ I ] , [ 4 ] ,[ 7 ] ,[S], [ 9 ] ,[lo]. Lemma 3. Let T,, T , E f ; ( M ) and X , X , , X 2 E P ( M ) , and let D , D,, D p , A be derivations on M , where A E F i ( M ) . Let R denote the curvature tensor field of the connection r. Then

317

360

A. J. LEDGER X. K. YANO

(12)

[X'L, T " ] = (V 1 T)" ,

(1 3)

[X:,X:l] = R(X1, X,)

__ -_

- --

,

(14)

[ X U ]=

(15)

[x:, xi1 = [ X , , X,lC.

VIA

+ [ X I , X?l" ,

Proof. Several equations can be proved by application of Corollary 1 . If p E M then r - ' ( p ) is a vector space, and has the structure of an abelian Lie group. If S E F ; ( M ) then S" is an invariant vector field on r - ] ( p ) and equation (9) follows immediately. We have, from Proposition 2,

[D,, D,]n = ( D I D , - D,D,)n = [ D , , D?]N. Since [D,?D,] is a derivation on M ,from Proposition 2 we have

[ D , , D,ln = [Q, D,ln

7

and hence equation (10)

[ D , T"]n = ( D ( n ( T ) )- ( D n ) ( T ) )

0

?r =

(4DT))

r,, equation

which gives equation (1 1 ) . Since X " = of (11). Since R(X,, X,)E T ; ( M ) , we have

7r

= (DT)YN)

9

(12) is a special case

[x:,x:]= [L 0 ,=~[L,, r , =~Rcx,Z2j+ rr,,,l?l , from which follows immediately equation ( 1 3).

[Xh,A]a = C,(An)

- A(Fyn) =

which gives equation (14). Since X c = of (10).

(V uA)n = (m>n ,

p,., equation (15) is

a special case

5. Almost complex structures We now consider the main problem, that is, to determine a class of tensor bundles which admit almost complex structures. For this purpose it is sufficient to consider contravariant tensor bundles since a Riemannian metric tensor field induces a fibre preserving diffeomorphism of T;M TrtSMM. Also --t

318

361

ALMOST COMPLEX STRUCTURES

the tangent bundle TIM of a Riemannian space always admits an almost complex structure [l], [ 5 ] . Hence we shall restrict attention to T'M, r > 1. Lemma 4. Let V and g be, respectively, a symmetric connection and a Riemannian metric tensor field on M , and E E P - ' ( M ) be nowhere zero on M . Then T'M admits three distributions which are mutually orthogonal with respect to a Riemannian metric tensor field 2 induced on T'M by V and g . Proof. For each p E M a scalar product <, > is defined on the vector space rr-'(p) by < T I , T , > = t,(T,), where, for any tensor T with components T i ~ i z " ' itr , is the covariant tensor associated to T by g. Thus t has components

tili2 . . . i r

- Tjlir'"i,gi -

.g. .

i2.i2 '

.

'

gi,j,.

7

where each repeated suffix indicates summation over its range. If S E T'M, then a scalar product, denoted by the same symbol <, >, is defined on the vector space (T'M),* by the three equations

(17)

< T,",T,"> = < T I ,T 2> < T " ,Xh > = 0 ,

(18)

= < X , , X ? >

(16)

0

n ,

o x ,

where X'l is the horizontal lift of X with respect to F . These equations are easily seen to determine on T'M with respect to which the horizontal distribution H , induced by r, is orthogonal to the fibres of T'M [3]. We now make use of E . For X E S I ( M ) , define the vertical lift X; of X with respect to E by

x;<= ( E 0X)l'

,

The map X X ; is then a monomorphism of P ( M ) J ' ( T ' M ) . Hence an n-dimensional C- vertical distribution V" is defined on T'M. Let V I be the distribution on T'M which is orthogonal to H and V s . Then H , L'/
4

~-&(p= ) { T :< T , E

0X >

= 0 for ull

X

Then L ' i = ( Y k ( p ) ) : . Let E l ( p ) be the sirbspuce of T r + ' ( p )defined by EL@)

Then 3 g ( p )

=

= {T:

< T , E > = O}

E L @ )0, F ' ( p ) .

319

,

E

S 1 ( p ) }.

362

A.

J. LEDGER 8: K. YANO

Proof. The first part of the lemma follows from the fact that the vertical lift preserves scalar products. To prove the second part it is sufficient to note that El-(p)0P ( p ) c J ; ( p ) , and dim ( E I ( p )0F ( p ) ) = n(n'-' - 1) = nr - n = dim F & ( p ) . Theorem. I f M admits an almost complex structure-and-a nowhere zero tensor field E E F T - l ( M ) ,then TrM admits an almost complex structure. Proof. Let F be an almost complex structure on M . We define a Cm tensor field J of type (1,l) on T'M by its action on the distributions H, T"$ and V L . Thus for X E Y ( M ) and T E P ( M ) define J by

J(X'l) = xi,J(X%) = - X " , J(T")= PJ,

(19)

where is obtained by contracting T 0F , and has components T*172. %iLF;r, where Tfi1?. and Ff are local components of T and F respectively. The restrictions of J to H f V L and V 1 are endomorphisms. and hence J is a tensor field on T'M. It is easily seen that J is C" and J? = - I , I being the unit tensor. Hence J is an almost complex structure on T'M. Corollary 12. Suppose a Riemannian manifold M admits an almost complex strirctirre. Then T'M admits an almost complex structure if (i) r is odd or (ii) r is even and M admits a nowhere zero vector field. Proof. ( i ) For r = 2s 1 choose E = (Og-')s, where g-' is the inverse of a metric tensor field g on M , and (Og-')sis the tensor product of g-I with itself s times. (ii) For r = 2s, s > 1, choose E = (Og-')"-'OX,where M is assumed to admit a nowhere zero vector field X . For Y = 2 choose E = X . 17

+

6. Integrability of the almost complex structure J

We now establish necessary and sufficient conditions for the integrability of J . Let e be the covariant tensor field of order r - 1 associated to E by g ; thus, with respect to local coordinates, e has components e ~ l i ~ , ,given . ~ ~ -by l eil~2...ir-l - gglj,gi2j

2...

.

~ i ~ - , . j ~ - , E ~ ~ ~ ~ " ' ~ r ~ ~

Proposition 3. Suppose M admits an almost complex structure F and a nowhere zero tensor field E E Y ' - ' ( M ) . Then the induced almost complex structure J is integrable if and only i f , f o r X , Y E F 1 ( M ) ,

R ( X , Y ) = 0 , TrE = 0 , P.rF = 0 , T.y--------- e

3 20

=0

,

363

ALMOST COMPLEX STRUCTURES

Proof. Let N be the Nijenhuis 2-form on T'M with values in F ( T r M ) , defined by

N(W1, W,) =

[Wit

W2l

+ J [ J W , , + J [ W i ,JW2l W2l

-

[ J W , ,JW2l

for W,, W , E P ( T r M ) . Then J is integrable if and only if N = 0. Suppose N = 0. Then for X , Y E P ( M ) ,N(X;., YY3)= 0 . Hence, putting W , = Xi,W 2 = Y ; we have, from (9), (12). (13), and the definition of J ,

R T X T ) = J(Pl.(E 0X))" - J(T.y(E O Y))" - [ X , Y ] "

since r has zero torsion. Now since E O F ( M ) is a subspace of .Yi(M) there is a unique T E P ( M ) orthogonal to this subspace and a unique Z E Y ' ( M ) such that

(V1,E)OX - ( P , E ) @ Y = T

+EOZ.

Then from (19) and (20) _ _

R ( X , Y ) = TI

-

Z" .

-~

Since R ( X , Y ) is vertical, 2'1= 0 and hence Z = 0. It follows from Corollary 11 that

(21) (22)

0,

R(X,Y ) T=O.

We thus have for all X , Y E Y ' ( M ) ,

( T I E )0Y =

(r,E ) 0X .

Since M is assumed to admit an almost complex structure, dim M 2 2. Hence by choosing X , Y to be linearly independent it follows that (23)

r,E = 0 .

We next consider the case N ( X g , T I ) = 0, where X i Then from (9), (12) and the definition of 1 we have (24)

It follows that

J ( ~ , T )=I

(r,T ) ' E V 1 . Choose

E

V' and TI

E

VI.

(r,Q .

T = S 0Y where S E Y r - I ( M ) , Y EF ( M )

321

364

A . J. LEDGER S: K . YANO

and < S, E > = 0 (since M is paracoinpact such an S exists and can be chosen to be non-zero in a neighbourhood of a point). Then by Lemma 5, TI'E V l and (24) imply that

( r , ~o ) FY t s o Fr,Y

=

( r , qo FY + s o r,(FY) .

Hence sB(r,F)Y

0 ,

and it follows immediately that (25)

T,F

1

0

.

Finally, from Lemma 5 the condition ( r , T ) ' \ E V l implies that 0 = e(T,S) = - ( r , e ) S

(26)

But S is any tensor field which satisfies (27)

where

I ,

S,

E

.

> = 0.

Hence we deduce that

r,e = n(X)e , (Y

E

F 1 ( M ) . Then

N I$

determined by

Thus (28)

N

= tl log e ( E ) = cl log

:<

E, E

>.

r

is the Riemannian connection associated with g then (23) implies (27) e and cy = 0.) Hence, from (27) and (28), the tensor field has zero covariant derivative, This proves the necessity of the conditions in Proposition 3 . To prove the sufficiency we note that (If

N ( X " , YV,)= N ( Y " , X h ) = JN(Y",, X " ) , N(X",, T") = JN(T", X " ) , N(Tp, T,")= 0

Thus N = 0 if N(X",, Y;) = N(X",, T") = 0.

,

Suppose TsE = 0 and

R ( X , Y ) = 0 for all X , Y E Y1(M). Then N ( X k , Y;) = - J [ X " , Y k ] - J [ X & , Y h ]- [X',, Y " ] = ( r y Y ) " - ( r , X ) h - [ X , Y]h = 0

322

.

365

ALMOST COMPLEX STRUCTURES

e = 0. The:i (27) follows and hence if T” E I/’ then (r.yT)i’E V l . If we next assume r,F = 0 then we have

Suppose V.y-

~

N ( X g , T ” )=

(r.yT)‘ - J(r.yT)v= 0 ,

which proves the sufficiency.

7. Kahlerian structure on T ’ M We now determine necessary and sufficient conditions for the metric 2 on T I M , defined in € j 5 , to be Kahlerian with respect to J . Proposition 4. g is Hermitian with respect to J if and only if < E , E > = 1 and g is Hermitian with respect to F . Proof. Suppose g is Hermitian with respect to J . Then for X , Y EF ( M ) ,

:X , y >

r

Y l X ’ t , y’I > = -1 JXZ., JYZ. i =

--

=
( . -

# ’

>

X>,Y k

~~T.

= 1 . Now let p E M and let S E Y-’ - ‘ ( p ) be non-zero such Hence ,E , E that < S , E > = 0. Then by Lemma 5 and the definition of J we have, for X , YE 7 l ( p ) ,

S,.S>>f X , Y ‘

.\%X,S@Y

x =

=

(S0X ) l , (S 8 Y ) ’

=

S O F X . S @ FY

’

J(S @

~

,T =

X)Ij.

.Y,S

r

J ( S @ Y ) ”>,

F X . FY

>

J

;r

.

Thus at p , < X , Y = F X , FY . . Since p is arbitrary, g is Hermitian with respect to F . The s u t k i e n c y of the above conditions is easily proved by the same method. Proposition 5. Slippose 2 is Hermitian with respect t o J . Then 2 is Kahlerian with respect t o J if und only if r is the Riemnrinian connection ussociatetl witli g , R = 0, T E = 0 unti T F : 0. Proof. Let N be the field of 2-forms on T r Mdefinedfor all W,,W, E 3 ’ ( T r M ) by n(W,. W,) = < W , , J W 2 >. Then 2 is Kiihlerian with respect to J if and only if N is closed and J is integrable [6, Chapter VII]. As usual it is sufficient to consider the action of N and d r y on the three distributions H , F‘>: and I/’ on T r M . Then for X , Y E F 1 ( M )and 7:, TI E V 1 we have

rU(XL, Y L ) = n(X’1, Y ” )= tu(T;‘,XYJ (29)

~(xl;, Y ” )= cu(Ty. Ty) =

*<

E@X,E @Y

< T , ,T , >

:z

,

323

= N(T:’,

X’t) = 0 ,

> -’ :: = < X , Y >

I

i~

,

366

A.

J. LEDGER & K. YANO

Suppose g is Kahlerian with respect to 1. Then by Propositions 3 and 4, R = 0, PsE = 0, and B,e = 0, for all X E Y ' ( M ) . Let p E M ,X E Y1(p), and choose T E F - l ( M ) such that < T , E > = 0 and < T , T > = 1 on some neighbourhood U of p . Since R = 0 parallel vector fields Y and Z exist on U with arbitrary initial values at p . Then using (9), (12) and Lemma 5 we have, on .-I@), 0 = &((T

0 Y ) " ,(T 0X ) " , X " )

< T @ Y , T @ F X > + < T 0FY, Bs(T 0Z ) > - < V.I(T 0Y ) , T 0FZ > = X < Y , F Z > + 2 < T,P.iT > < F Y , Z > + < F Y , r,z > - < B ~ YFZ , > = ( P , p ) ( Y ,FZ) - 2 < T , P \ T > < Y , FZ > . =X

Since F is non-singular it follows that Bsg = n(X)g

9

for some a E ,TI@), Then since P,E = 0 and Bse = 0 it follows easily that for all X E F ( p ) , 0 = Vse = ( r - l)cu(X)e

.

The tensor e is non-zero and so n = 0. Thus Vg = 0 at p and hence on M since p is arbitrary, It follows that P, having no torsion, is the Riemannian connection associated with g . We now prove the sufficiency of the above conditions by showing that the 2-form n is exact. Let X E Y1(M), and Ti' E V l , Define a 1-form?! , on TrM as follows: at each point S E T r M , P(X") = ,

j ( X ; ) = 0, /?(T")= & < S , T >

.

Then using (29) it follows after some calculation that LY = d p . Hence d n = 0, and this together with Proposition 3 proves the sufficiency. 8. Integrability of H Proposition 6. H

+ V" and H + Y-L

+ V Eis integrable if and only i f R = 0 and for X Ep ( M ) ,

P x E = a ( X ) E , where n ( X ) = < E , V s E > * Proof. It follows from (12) and (13) that H tion if and only if for X , , X , E Y1(M), (31)

f

(BXI(E0XJ)" E V's ,

3 24

V Eis an integrable distribu-

367

ALMOST COMPLEX STRUCTURES

R ( X , , X,)

E

VE

.

Let Y , and Y , be orthogonal vectors at p c M , and let < T , E > = 0 at p . Then from (16), (32) and Corollary 6, 0 =
0Y , >

= < T , T >

.

Hence R ( X , , X , ) Y , = cY, where c is some real number which depends on X , and X , . Since Y , is arbitrary -~ it follows that R ( X , , X , ) = cl at p . Then at any point S E n-’(p> we have R ( X , , X , ) = -crS”, and by choosing S” E V 1 it follows that R ( X , , X , ) = 0 at S; hence c = 0. Since p , X , and X , are arbitrary we have R = 0 on M . Using (30) and Lemma 5 we obtain FAYE = n ( X ) E and N is then uniquely determined by this equation. The proof of the sufficiency is immediate, Proposition 7. H V 1 is integrable if and only if R = 0 uncf for X EP ( M ) , < e , Pse > V,ye = a ( X ) e , where LY = _____

+



Proof. The proof is similar to that of Proposition 6 and we shall use the same notation. It follows from (12), (13) and Lemma 5 that H + V-L is an integrable distribution if and only if for S“E V l ,

(r,l(s

(33)

oX N

E v1

,

R(X,,X?) E V l .

(34)

then from (16), (34) and Corollary 6, 0 = < R ( X , , X,)(E 0 Y , ) , E

0Y , >

= < E , E > < R ( X , , X 2 ) Y , ,Y , >

.

Hence, as before, R = 0. From (33) we obtain

o = < x , ,Y > for Y E P ( p ) . Hence

o =

=

e(r,,s) = -(rlle)s.

It follows that r , , l e = n ( X , ) e at p . Since p and X , are arbitrary we obtain r , , e = cu(X,)e on M , and N is then uniquely determined, The proof of the sufficiency is immediate.

325

368

A. Y. LEDGER S. K. YANO

References P. Dombrowski, On the goenietry of tarigent brtridles, J. Reine Angew. Math. 210 (1962) 73-88. S. Helgason, Diflerential geometry arid symmetric sprrces, Academic Press, New York, 1962. S.Sasaki, On the differentialgeometry of tarigerzt brindles of Rierrimriirrrz r~ioiiifolds, TBhoku Math. J. 10 (1958) 338-354. A. J. Ledger & K. Yano, The tnrzgerit brrridle of a locally syrnnietric space, J. London Math. SOC.40 (1965) 487-492. S. Tachibana & M. Okumura, 011the rrlniost complex strrrctrrrc of tangent brrridles of Riemarirzicrri spaces, TBhoku Math. J . 14 (1962) 158-161. K. Yano, Diflereritirrl geometry on corriplex arid rrlmost complex spaces, Pergamon, New York, 1965. K. Yano & S . Ishihara, Almost complex strrrctrrrrs iritlrrced i r r torigeril brindles, K6dai Math. Sem. Rep. 19 (1967) 1-27. K. Yano & S . Kobayashi, Puolorigatiaris of terisor fields arid coririectiorrs to trrrrgerit brirrdles I, J. Math. SOC.Japan 18 (1966) 194-210. K. Yano & A. J. Ledger, Lirieor eoririectioris ori trrngerzt birridleJ, J. London Math. SOC.39 (1964) 495-500. K . Yano & E. M. Patterson, Vertical aritl corripletc lifts from ( I m~riifolr/ta its cutrrrigerit brrridle, J. Math. SOC.Japan 19 (1967) 91-1 13.

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