space. A plane section r is called holomorphic (respectively anti-holomorphic) ... curvatures o] M are constant and i] dim M > 3, then M is a complex space ]orm.
SOME CHARACTERIZATIONS OF COMPLEX SPACE FORMS BANG-YEN CHEN AND KOICHI OGIUE 1. Introduction. Let M be a Kaehler manifold with complex structure J and Riemann metric g. By a plane section we mean a 2-dimensional linear subspace of a tangent space. A plane section r is called holomorphic (respectively anti-holomorphic) r (respectively J is perpendicular to r). The sectional curvature if J for a holomorphic (respectively anti-holomorphic) plane section is called holomorphic (respectively anti-holomorphic) sectional curvature. A Kaehler manifold of constant holomorphic sectional curvature is called a complex space ]orm. It is well known that a complex space form has constant anti-holomorphic sectional curvatures. Conversely, in Section 3 we shall prove the following theorem. THEOREM 1. Let M be a Kaehler mani]old. I] the anti-holomorphic sectional curvatures o] M are constant and i] dim M > 3, then M is a complex space ]orm. A Kaehler manifold M is said to satisfy the axiom o] holomorphic planes (respectively the axiom of anti-holomorphic planes) if for each x M and each holomorphic (respectively anti-holomorphic) plane r, there exists a 2-dimensional totally geodesic submanifold N such that x N and T(N) -. Yano and Mogi [4] proved that a Kaehler manifold with the axiom of holomorphic planes is a complex space form. In Section 4 we shall prove the following theorem.
THEOREM 2. Let M be a Kaehler mani]old. I] M satisfies the axiom o] antiholomorphic planes and i] dim M > 3, then M is a complex space ]orm. 2. Preliminaries.
In this section we shall give a brief summary of basic
formulae.
Let M be a Kaehler manifold with complex structure J and Riemann metric g. We denote by R the curvature tensor field of M. Then we have
(2.1) R(JX, JY) R(X, Y) (2.2) R(X, Y)JZ JR(X, Y)Z. Let K(X, Y) be the sectional curvature of M determined by orthonormal vectors X and Y. Then we have (2.3) K(JX, JY)= K(X, Y) Received April 16, 1973. The first author was supported in part by NSF grant GP-36684. The second author was supported in part by the Matsunaga Science Foundation.
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BANG-YEN CHEN AND KOICHI OGIUE
K(X, JY)= K(JX, Y).
(2.4)
The following is easily seen.
(2.5) Orthonormal vectors X and Y span an anti-holomorphic section if and only if X, Y and JX are orthonormal. Moreover we have K(X, Y) + K(X, JY) provided that X and Y (2.6) g(R(X, JX)JY, Y) span an anti-holomorphic section.
By H(X) we denote the holomorphic sectional curvature determined by X, i.e., H(X) K(X, JX). Let N be a submanifold of M, and let and V be the covariant differentiations on M and N respectively. Then the second fundamental form of the immersion is defined by z(X, Y) V xY x Y, where X and Y are vector fields tangent to N. is a normal bundle valued symmetric 2-form on N. For a vector field normal to N we write V x -A X q- Dx, where -A X (respectively D x() denotes the tangential (respectively normal) component of x}. Since R(X, Y)} VxVr} VrVx Vtx,r)}, we can deduce that
(2.7) R(X, Y)
(.A)X
(xA)Y (modulo normal component).
3. Proof of Theorem 1. Assume that the anti-holomorphic sectional curvature of M is a constant c. Let X and Y be orthonormal vectors which span an anti-holomorphic section. Then it is easily seen from (2.5) that X JY span an antiY and JX holomorphic section. Therefore, using (2.1), (2.2) and (2.6), we have
c
K(X
+
Y, JX- J Y)
Since K(X, Y)
K(X, JY)
g
/
1/4[H(X)
+ H(Y)
/
2{g(x, Y)
/
/
+ g(x, JY)}].
c, we have
H(X) zr- H(Y) 8c. Let x be an arbitrary point of M and let U and V be arbitrary unit vectors in Tx(M). Since dim M _> 3, then we can choose a unit vector W in U, JU} } and {{V, JV}} where {/"" }} denotes the plane section spanned by _1_ denotes the orthogonal complement in Tx(M). It is clear that [{U, W}} and I/V, W}} are anti-holomorphic sections. Therefore from (3.1) we have 8c and H(V) + H(W) H(V). 8c, which imply H(U) H(U) + H(W) Since U and V are arbitrary, the holomorphic sectional curvature does not depend on the choice of a holomorphic section at x M. Since x is arbitrary, the complex version of a well-known theorem of F. Schur (cf., for example, [2; Theorem 7.5 in Chapter IX]) implies that M is complex space form.
(3.1)
-,
COMPLEX SPACE FORMS
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Let x be an arbitrary point of M and let X and Y be arbitrary orthonormal vectors in Tx(M) which span an anti-holomorphic section r. Let N be a totally geodesic submanifold such that x N and Tx(N) r. Since r is anti-holomorphic, JX is normal to N. Therefore from (2.7) we have R(X, Y)JX =- 0 (modulo normal component). In particular O. we have g(R(X, Y)JX, X) Now our assertion follows from the following lemma. 0 for every orthonormal X, Y, JX LEMMA [3]. If g(R(X, Y)JX, X) T(M) and for every point x of M, then M is a complex space form. Proof. Let X be an arbitrary point of M and let X and Y be arbitrary 4. Proof of Theorem 2.
orthonormal vectors at x which span an anti-holomorphic section. X Y and JX JY span an anti-holomorphic section so that
Then
from whieh we can deduce the following statement. H(X) H(Y) provided tha X, Y and JX are orhonormal. (4.1) Let U and V be arbitrary unit vectors in T(M). If U, V is a holomorphie section, then it is clear that H(U) H(V). If l/U, V}/is not a holomorphie section, then, since dim M >_ a, we can choose a unit vector W in {U, JU} +/IV, JV} }-. Therefore (4:.1) implies
H(U)
H(W)
H(V).
Since U and V are arbitrary, the holomorphic sectional curvature does not depend on the choice of a holomorphic section at x M. Since x is arbitrary, the complex version of a well-known theorem of F. Schur implies that M is a complex space form. REFERENCES 1. E. CARTAN, Lecons sur la Gomtrie des Espaces de Riemann, Paris, Gauthier-Villars, 1946. 2. S. KOBAYASHI AND K. NOMIZU, Foundations of Differential Geometry, Vol. II, New York, Interscience, 1969. 3. KOICHI OGIUE, On inuariant immersions, Ann. Mat. Ptr Appl., vol. 80(1968), pp. 387-397. 4. KENTARO YANO AND ISAMU MOGI, On real representation of Kaehlerian manifolds, Aan. of Math., vol. 61(1955), pp. 170-189.
DEPARTMENT OF MATHEMATICS MICHIGAN STATE UNIVERSITY EAST LANSING MICHIGAN 48823
