| Abstract: | The Properties of submanifolds in a Bochner-Kaehler manifold have been studied
mainly in the cases that the submanifolds are totally real by Yano, K., Houh, 0. S. and others.
The main purpose of the present paper is to study whether the condition for the
submanifold to be totolly real in their theorems is necessary, and to prove some
theorems which are analogous to those mentioned above.
A submanifold M^n of Kaehlerian manifold M^2m is called totally real or antiinvariant,if each tangent space of M^n is mapped into the normal space by the complex structure $\{F_{\nu \mu }}\]$ of M^2m. Similarly, a submanifold M^n of Kaehlerian manifold M^2m is called anti-in variant with respect to L', if each tangent space of M^n is mapped into the normal space by the operator L' of M^2m.
We obtain:
(1) A necessary and sufficient condition for a totally umbilical submanifold M^n,
n>3, in a Boohner-Kaehler manifold M^2m to be conformally flat is that the submanifold M^n is either a totally real submanifold or an anti-invariant submanifold with respect to L'.
(2) Let M^n be the submanifold immersed in a Boohner-Kaehler manifold M^2m.
If each tangent vector of M^n is Ricci principal direction and Ricci principal curvature $\{\rho _h}\]$ does not equal $\frac{{\tilde K}}{{4(m + 1)}}\]$ , then the anti-invariant submanifold with respect to L^' coincides with the totally real submanifold.
(3) Let M^n be a totally umbilical submanifold immersed in a Boohner-Kaehler
manifold M^2m If M^n is a totally real submanifold or an anti-invariant submanifold,then the sectional curvature of Mn is given by
$\rho (u,v) = \frac{1}{8}(\tilde K(u) + \tilde K(v)) + \sum\limits_{x = n + 1}^{2m} {{H^2}} ({e_x})\]$(A)
where H(e_x) =H_x. Conversely, if the sectional curvature of M^n satisfying the condition mentioned in (2) is given by (A) for any two orthonormal tangent vectors u^\alpha and $v^\alpha$ then M^n is a totally real submanifold. |
