ANTI-INVARIANT SUBMANIFOLDS IN A BOCHNER-KAEHLER MANIFOLD

The Properties of submanifolds in a Bochner-Kaehler manifold have been studied mainly in the cases that the submanifolds are totally real by Vane, K., Houh, C. 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_(νμ) of M~(2m). Similarly, a submanifold M~n of Kaehlerian manifold M~(2m) is called anti-invariant 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, n3, in a Bochner-Kachler manifold M~(2m) to be confromally fiat is that the submanifold M~n is either a totally real submanifold or an anti-ivariant submanifold with respect to L′.(2) Let M~n be the submanifold immersed in a Bochner-Kaehler manifold M~(2m). If each tangent vector of M~n is Ricci principal direction and Ricci principal curvature ρ_h does not equal (?)/(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 Bochner-Kaehler manifold M~(2m). If M~n is a totally real submanifold or an anti-invariant submanifold, then the sectional curvature of M~n is given byρ(u,v)=1/8((?)(u)+(?)(v))+sum from x=n+1 to 2m (H~2((?)_x)), (A) where H((?)_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~a and v~a, then M~n is a totally real submanifold.