Journal of Geometry Vol. 49 (1994)

0047-2468/94/020106-it51.50+0.20/0 (c) 1994 Birkh~iuser Vertag, Basel

H O L O M O R P H I C A L L Y P R O J E C T I V E T R A N S F O R M A T I O N S ON

C O M P L E X R I E M A N N I A N M A N I F O L D

Dedica ted to N.K, S tephanid is on the occasion of his 65 th bi r thday.

Stefan Ivanov

The aim of the paper is to prove that if a complex Riemannian manifold with holomorphic characteristic connection is holomorphically projective equivalent to a locally symmetric space then it is a complex Riemannian manifold of pointwise constant holomorphic characteristic sectional curvature.

0. I N T R O D U C T I O N

According to the Beitrami theorem on classical differential geometry there is a geodesic map-

ping of a Riemannian manifold M(dim M > 2) onto a manifold of constant curvature iff M

is of constant curvature.

In the geometry of complex manifold we can consider instead of geodesic mappings, holomor-

phically projective transformations [2], [5], [8], [10]. In [6] M.Prvanovic gives an analogous

of the Beltrami theorem in the Kaehler and hyperbolic Kaehler cases.

Generalizing Beltrami theorem N.S. Sinjukov [7] proved that there is a geodesic mapping of

a Riemannian manifold M(dim M >_ 3) onto locally symmetric manifold iff M is of constant

sectional curvatures.

An analogous to the Sinjukov theorem in the Kaehler and hyperbolic Kaehler cases was

given by M. Prvanovic in [6]. She has proved that there is holomorphically projective trans-

formations of a Kaehler (hyperbolic Kaehler) manifold M onto a locally symmetric space iff

M is the space of constant holomorphic sectional curvatures (the space of almost constant

holomorphic sectional curvatures).

Ivanov 107

In this paper we find an analogous to the Beltrami and Sinjukov theorems in geometry of

complex Pdemannian manifold. The aim of the paper is to prove the following

T H E O R E M A. Let M be an n-dimensional (n >_ 3) complex Riemannian manifold with

non fiat holomorphic characteristic connection. Then M is holomorphicaIIy projective equiv-

alent to a locally symmetric space iff either

a) M is locally holomorphicaI isometric to the complex sphere C S ~ in C ' ;

O F

b) M is locally comformal equivalent to the complex sphere C S ~ in C ~ or to the fiat space C n ,

1. C O M P L E X R I E M A N N I A N M A N I F O L D .

Let M be an n-dimensional complex manifold. We denote by (M, J ) the manifold considered

as a real 2n-dimensional space with the induced complex structure J. The tangential space to

(M, J ) at a point p C M and its complexification are denoted by TpM and T i M respectively.

The algebras of real differentiable vector fields, complex differentiable vector fields, vector

fields of type (1, 0) and vector fields of type (0, 1) on M are denoted by X M , XCM, X I ' ~

and X ~ respectively.

If z l , . . . , z n are holomorphic coordinate functions in a coordinate neighbourhood U C M

and z ~ = x ~ + x/Z-Ty ~ , c~ = 1, 2 , . . . , n , then the complex vector fields

1 [o/oxo _ ,FfOlOyO] 0~ := Z~ = OlOz ~ = -~

{ ~ := = = Z~ O/ Oz -~

form a bases for Xl '~176

A complex Riemannian metric on a complex Riemannian manifold M is a covariant sym-

metric 2-tensor field G : X ~ x XCM -+ C, which is nondegenerate and

w) = C( z, W), Z, W e X~

G ( Z , W ) = O, Z , W e XI '~

where the symbol 'bar ' denote complex conjugation.

Unless otherwise s ta ted , Greek indices cz, r 7... run from i to n, while latin capitals A, B, C

... run through 1, ...n,]-, . . . , g . In terms of local holomorphic coordinates z 1, ..., z ~ we set

GAB = G(Z~, ZB)

108 tvanov

Then the defining conditions for a complex Riemannian metric become

Gaf f = G ~ ; G o a = G 5 o = 0

The fundamental tensor q~ on complex Riemannian manifold is defined by [1]

~z ,~ = 0~ ,G~p,

The Lie form 0 is defined by

~7fi,7 = r ~ , ~ = ~a ,7 = 0.

where {G ~'r} is the inverse matrix to the matrix {G~-~}.

There are four classes of complex Riemannian manifold with respect to the fundamental

tensor r

1.(I) = 0 i.e. G ~ are holomorphic functions.This is the class of complex analytic Pdemannian

manifolds introduced by R. Penrous [4], s.f. [3].

2.~.y ,5 = O-~G;~.y . This is the class of complex Riemannian manifolds locally conformal to

the complex analytic Riemannian manifold.

3 . 0 = 0 .

4. No conditions.

Let F4cB are the local components of the Levi-Civita connection V for the metric G with

respect to the holomorphic coordinate system. The local components of the unique charac-

teristic connection D defined in [1] are given by:

D~p 7 D~---= "~" = P ~ , ~ D~o,

D r ~ = D ~ - = D " y - = D 2 - = v -s~ = D , ~ ~ ~,~ ,,~ D-5~ = O.

In [1] it is shown that the characteristic connection D on complex Riemannian manifold is

the unique torsion-free almost complex ( D J = 0) connection with the property

This condition implies

DC~AB = OAB,C

The characteristic connection D is holomorphic if D x Y is holomorphic vector field whenever

X, Y are holomorphic vector fields. Thus D is holomorphic connection iff the local compo-

nents D r ~Z are holomorphic functions.

Let K be the characteristic curvature tensor i.e. the curvature tensor of the characteristic

Ivanov 109

connection D. The characteristic curvature tensor of type (0,4) is defined by K(X, Y, Z, W) =

G ( K ( X , Y ) Z , W ) . The local components of K are given by K2.c = K(ZA, Z , ) Z c and

K A B C E = GDEKDBc . The nonzero components of K are given by [1]:

= O~D~,~ D ~ ~ ~ ~ ~ �9 K ~ K,m. ~ - O~ ~.y + D ~ D ~.~ - D ~ D ~.y, ~Z'Y = K ~.y.

~ K ~ -

It is clear that K~0 ~ are zero iff D is holomorphic connection.

There are two Ricci-type contractions: p ~ = K2~z, s ~ = I ( ~ . The scalar curvature T

is defined by r = G~p~ z.

Every nondegenerate 2-pla~e in X pMa,~ is said to be holomorphic 2-plane. Holomorphic

characteristic sectional curvature for a holomorphic 2-plane E = span {Z, W},

Z , W E X~'~ C M is defined by

K~( E) = K ( Z, W, Z, W) a( z , z ) c ( w , w ) - ( G( Z, W) )~

In [1] it is proved that a complex Riemannian manifold is of pointwise constant holomorphic

characteristic sectional curvatures c iff

where c = u+x/'-Zlv is a differentiable function on M.

We need the following classification theorem

T H E O R E M I [1]. Let (M, G) be an n-dimensional (n >_ 3) complex JRiemannian manifold

with holomorphic characteristic connection D of pointwise constant holomorphic character-

istic sectional curvature c which is not identically zero. Then

i) If c is constant then (M, G) is complex analytic Riemannian manifold locally hoIomorphi-

cal isometric to the complex sphere in C~;

ii) If c is not constant then (M, G) is locally conformal equivalent to the unit complex sphere in C n.

2. H O L O M O R P H I C A L L Y P R O J E C T I V E T R A N S F O R M A T I O N S ON

C O M P L E X R I E M A N N I A N M A N I F O L D

Two torsion-free almost complex linear connections V and V' on complex manifold (M, J)

are said to be holomorphically projective (H-projective) equivalent if they have common

holomorphic planar curves [2], [8], [10] i.e they are connected in the following way

(2.1) VScY = V x Y + ~ [q(X)Y + q (Y )X - q ( J X ) J Y - q ( J Y ) J X ] ,

1i0 Ivanov

where q is a 1-form on M and X, Y C WM.

P R O P O S I T I O N 2.1. Let (M,G,D) and (M,G 'D ' ) be two complex Riemannian manifolds

with holomorphic characteristic connections D and D' which are H-projective equivalent as

(2.1). Then the 1-form q is hoIomorphic and O-closed.

Proof: In holomorphic coordinates from (2.1) we have

'z = D ~ +q~5~+q~5~, (2.2) D~p o,#

s~/ 3, where 5 denote the Kroneker symbol. Since D~# and D~# are holomorphic functions we

obtain

which implies

0-2q~ = 0.

To prove that q is 0-closed we have to compute the curvature tensors K and K ~ for D and

D' respectively. From (2.2) we have

(2.3) [Lo K ~ Z ~ = K ~ + L~5~ - - -

where

(2.4) Lz.y = D~q~ - q~qz

From (2.3) for the tensors p ~ and Pap we compute

t p ~ = p~Z + nLr - L ~

Since the tensors Pip and p~z are symmetric [1] we obtain

(n + - = 0

From (2.4) we derive O~q~ - O~q~ = 0 which implies Oq = O. Q . E . D .

T H E O R E M 2.1. Let (M, G, D) be an n-dimensional (n >_ 3) complex [~iemannian mani-

fold with hoIomorphic characteristic connection D. Then M is H-projective fiat iff M is of

pointwise constant holomorphic characteristic sectional curvatures.

Proof: The theorem of Y.Tashiro [8] states that D is H-project ive fiat iff its holomorphic

projective tensor H P ( K ) is zero. In local holomorphic coordinates the condition H P ( K ) = 0

can be writ ten in the following way:

1 (2.5) t ( , ~ -- - [pz,vG~;, - p~,rGz;,]"

n 1

Ivanov 111

1

Since D is holomorphic connection then I i ~ a = 0. From (2.5) and (2.6) by a contraction

we obtain

(2.7)

Substituting (2.7) in (2.5) we get

T

p~z = nG~Z; s~z = 0.

T

which implies the assertion. Q . E . D .

Since every two projectively flat connections are H-projectively equivalent from Theorem

2.! we derive

T H E O R E M 2.2. ( H O L O M O R P H I C B E L T R A M I T H E O R E M ) . There is a holo-

morphically projective mapping of a complex Riemannian manifold M ( d i m M = n _>3) with

holomorphie characteristic connection onto a complex Riemannian manifold with pointwise

constant holomorphic characteristic curvatures iff M is of pointwise constant holomorphic

characteristic curvatures.

DEFINITION. A complex Riemannian manifold M is said to be characteristic Einstein if

P~o = fG~o, s ~ = O,

where f = u + xfL-fv is a smooth function on M.

P R O P O S I T I O N 2.3.Let (M, G) be an n-dimensional (n > 3) complex Riemannian char-

acteristic Einstein manifold. Then the characteristic scalar curvature r is anti-hoIomorphic

function and p~z = ~G=z.

Proof: From the second Bianchi identity we have

By a contraction we obtain

Since D.~Go~ = 0 we get

~R~.~u + D ~ R ~ , + D~R~p~ = O.

D~p~u + D ~ R ~ u - D~p~u = O.

DaTGTu + D p R ~ , - DTTG~, = O.

Multiplying with G 7" we have (n - 2 ) O j = 0 which implies r is anti-holomorphic function.

Q . E . D .

112 Nanov

P R O P O S I T I O N 2.4. Let (M,G) be an n-dimensional (n >_ 3) complex R~emannian

characteristic Einstein manifold with hoIomorphic characteristic connection and the charac-

teristic scalar curvature r is not identically zero. Then:

i)T is constant iff (M, G) is complex analytic Riemannian manifold;

ii) if ~" is not constant (M, G) is locally conformal equivalent to a complex analytic Ricman-

nian manifold;

Proof: Since D is holomorphic connection the functions p,~ are holomorphic functions and

we h a v e

0 = ~ p ~ = 0 ~ - G ~ + ~-~C,o.~. Since ~- # 0 we get ~ , ~ = O-~Log('c)G~.y which implies ii) (see [1]). It is clear that o~" = 0

iff ~.y,~ = 0 ,i.e. T is holomorphic function iff (M, G) is complex analytic Riemannian man-

ifold. From Proposition 2.3 0~- = 0 and ~- is holomorphic function iff ~- is constant. Q.E.D.

Proposition 2.3 and Proposition 2.4 are analogous in the geometry of complex Riemannian

manifold to the Herglotz theorem.

T H E O R E M 2.5. Let (M,G) be an n-dimensional (n >_ 3) complex Riemannian ch, ar-

acteristic Einstein manifold. Then M is if-projective fiat iff M is of pointwise constant

holomorphic characteristic curvatures.

Proof," We consider the condition t I P ( K ) = 0, where H P ( K ) denote the holomorphic pro-

jective tensor of the characteristic curvature tensor K. Since s ~ = 0 from (2.6) we have

K ~ - ~ = 0 which implies the characteristic connection D is holomorphic. Now the theorem

follows from Theorem 2.1. Q.E.D.

T H E O R E M 2.6. Let (M, G, D) be an n-dimensional (n > 3) complex l{iemannian mani-

fold with holomorphic characteristic connection D. If there exists holomorphically projective

transformation onto a locally symmetric complex Riemannian manifold (M,G'D') with holo-

morphic connection D',then ( M , G , D ) is of pointwise constant holomorphic characteristic

curvatures.

Proof: Let q be 1-form on M and

(2.8) D ~ = D:~ + q~5~ + q~5~,

with respect to a holomorphic coordinate system z l . . . zn.From Proposition 2.1 the 1-form

q is holomorphic and 0-closed. First we prove the following

L E M M A 1. By the conditions of the Theorem 2.6 either M is characteristic Einstein or

the 1-form q ~atisfies the following condition:

Ivanov 113

where f is smooth function on M.

Proof of the Lemma 1. For the non zero components of the characteristic curvature tensors

K and K' for D and D ~ respectively we have:

K~Z ~ = K ~ - LZ~8 ~ + I ( ~ = K~.~ (2.9)

where

(2J0) L ~ = D~q~ - q~qz

The 1-form q is 0-closed which implies L,Z = L ~ . Since D' is locally symmetric we have

(2.11) n ' w,E ~ A X X B C D -= O.

This equality can be written in the following form

(2.12) n ' I/,,x n ' ~ ' ~ '),

From (2.11) and the Ricci identities we have

(2.13) I / , s I / I F I j I S I / I F • I'z-,S I / I F I / I F I / I S ~ x A B C J ~ S D E "Jc ~ A B D ~ * C S F ~ ~ A B E ~ C D S - - , ~ A B S ~ C D E -= O.

From (2.13) we derive

Using (2.9) we replace K ' by K:

(2.15) L~K~,o + L~,I~L, + L#c~,,- L~,~,~

a ) , +(L ,~K;~ , + L,~K:p, )6 ~ - (L~K:~.~ + L , ~ K ~ , ) ~ , : 0.

Fnrther we follow the notation in [7], [9]. We denote by A ~ the following tensor:

L (2.16) A , ~ = L~p~,~ - G~ ~p~,

where p~ := G~"p,~. Multiplying (2.15) by G~,~G ~'~ we obtain:

where

114 Ivanov

Then the tensor B , , ~ defined by

(2.17) B~..,,~ = A , ~ + A ~ , , ,

is skew-symmetric in v, n and #, a .

Transvecting B , ~ o + B ~ , = 0 with G "~ we get

~ -G L ~ ~ (n + l )L~p~ - L,~p~ = Lp,~ + rL,~ ~ ~p~, (2.18)

where

L = G ~ L ~ ,

Since L is symmetric from (2.18) we obtain

L~ = L~G ~.

(2.19) r r = L

Substituting (2.19) in (2.16) from (2.17) we get

These algebraic properties imply

(2.20)

Contracting (2.18) with G ~" we have

Now (2.18) entails

where

B g u ~ a : 0

~ LT L~p~ = --

n

c~

nL.~p, : LE,~g + 7L,~,

T

is the characteristic Einstein tensor. Replacing this expression in (2.20) we get

where Y

The last equation mean that either E,~ = 0 or N,~ = 0 which proves the Lemma i.

We continue the proof of Theorem 2.6. The holomorphic projective tensor HP(IC) of the

connection D' has the form

_ : A (2.21) H P ( K ),,~'r ~'~p-r n 1

Ivanov 115

where ~ fx p ~ = K),~.y.

From (2.12) we obtain D'~p;^~ = 0. Combining with (2.21) we get

(2.22) D ' [ H P ( K ' ) ] ~ = o.

Since the holomorphic projective tensor is invariant under the H-projective transformations

([8],[21,[10]),(2.22) implies

(2.23) D'~ [HP(K)I~p~ = 0,

where the holomorphic projective tensor H P ( K ) of the connection D has the form (2.21).

Using (2.9) from (2.23) we obtain

(2.24) D, [HP(K)]2~.y = 2q, HP(K)2z.~

+q~HP(-K)x,r + q~HP(I()~u. Y + q.~HP(I()~r - cl~ H P (K)~p.~S,.~ x

By Lemma 1 we have to consider two cases:

CASE A. T

peztz = ~ G ~ # .

Then the holomorphic projective tensor H P ( K ) of type (0,4) has the form

T (2.25) HP(I<)~ ,~ = K ~ ~(~ - 1) [Gp~ao~ - ao~G,~]

Since K ~ . x = - iu (see [1]), from (2.2,5) it follows

(2.26) HP(K)~p,~ = - H P ( K ) o , ~ ,

Since D is holomorphic characteristic connection, lowering index A in (2.24),then symmetriz-

ing with respect to A,7 and taking into account (2.26) we obtain

(2.27) q~HP(K)~z.rG,) , + q,,I-IP(K)tp),G,, r

-q.yH P( K ) ~ , ~ - q~ H P( K)~o,. ~ = O.

Transvecting with respect to # and A we obtain q~HP(K)~o. ~ = 0 and substituting in (2.27)

we get

(2.28) q.rHP(K)~zza + q:~HP(I()~Z,. ~ = 0

Since q is not identically zero (2.28) implies H P ( K ) = 0 and from (2.25) (M,G) is of

pointwise constant holomorphic characteristic curvatures.

116 Ivanov

CASE B.

Substituting this equality in (2.9) we get

This tensor satisfies all the conditions of the forms (2.22), (2.24), (2.26) and (2.28). In She i3, same way as above we can conclude K~S~ = 0 which implies (M, G) is of pointwise constmrtt

holomorphic characteristic curvature. This completes the proof of the theorem. Q . E . D .

Proof of the theorem A. Combining Theorem 2.6 and Theorem I we get the proof of Theorem

A.

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[5] PRVANOVIC M. , Holomorphically projective transformations in a locally product space, Math.Balkanika 1 (1971) 195 - 213.

[6] PRVANOVIC M. , A note of the holomorphically projective transformations of the KaehIer spaces. , Tensor N.S. 35 (1981) 99 - 104.

[7] SINJUKOV N.S., Geodesic mapping onto a symmetric spaces , Dokl. Akad. Nauk USSR 98 (1954) 21 - 23.

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University of Sofia, Faculty of Mathematics and Informatics, Department of Geometry, bul. James Bouchier 5, 1126 Sofia , BULGARIA

Eingegangen am 20. Juni 1991