Kaehler submanifolds with the second fundamental form satisfying certain conditions

Yoshio Matsuyama
Let M n be a Kaehler submanifold in a complex space form and x any point of M n. Then there exists a neighborhood U of x which a local field ξ of any normal vector and the second fundamental form A ξ in the direction of ξ are defined on U. In the present paper we will give a characterization of a Kaehler submanifold in a real space form which satisfies (R(X, Y)A ξ)Z = 0 for all vectors X, Y and Z tangent to M n and A ξ in the direction of any normal ξ, where R is the curvature tensor of M n which is defined on U. M.S.C. 2000: 53C40, 53B25.