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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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[3] LEBRUN C . , Spaces of complex null-geodesics in complex Riemannian geometry. , TAMS 2781 (1983) 209 - 231.
[4] PENROUSE R. , Non-linear gravitons and curved twistor theory, Gem Relativity and Gray. 7 (1976) 31 - 52.
[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.
[8] TASHIRO Y. , On holomorphically projective correspondence in an almost complex space, Math J. Okayama Univ. 61957N2147 - 152.
[9] VENZI P., On geodesic mappings in Riemannian and pseudo Riemannian manifold, Tensor N.S. 32 (1978) 193 - 198.
[10] YANO K. ,Differential Geometry on Complex and Almost Complex Spaces, Pergamon Press 1965.
University of Sofia, Faculty of Mathematics and Informatics, Department of Geometry, bul. James Bouchier 5, 1126 Sofia , BULGARIA
Eingegangen am 20. Juni 1991
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