On totally real submanifolds

Abstract

The following results are proved: Theorem 1. A totally real semiparallel submanifold of constant curvature with parallel f-structure in the normal bundle of a K\"ahler manifold N is flat or a totally geodesic submanifold of N. Theorem 2. A totally real minimal semiparallel submanifold M with parallel f-structure in the normal bundle and of constant length of the second fundamental form (or equivalently of constant scalar curvature) of a complex space form N is totally geodesic in N or of positive scalar curvature. Moreover, if the scalar curvature of M vanishes, then M is flat. Theorem 3. A complete, compact totally real submanifold with parallel mean curvature vector, parallel f-structure in the normal bundle and commutative second fundamental forms of a simply connected complete complex space form is totally geodesic or a pythagorean product of circles. Note that if M is a totally real submanifold of a K\"ahler manifold N and the dimension of N is twice the dimension of M, then the f-structure in the normal bundle vanishes.