On submanifolds satisfying chen's equality in a real space form

Abstract

Einstein, conformally flat, semisymmetric, and Ricci-semisymmetric submanifolds satisfying Chen's equality in a real space form are studied. We prove that an n-dimensional (n >= 3) submanifold of a real space form (M) over tilde (n+m)(c) satisfying Chen's equality is (i) Einstein if and only if it is a totally geodesic submanifold of constant curvature c; and (ii) conformally flat if and only if inf K=c, where K denotes the sectional curvatures of the submanifold. We also classify semisymmetric and Ricci-semisymmetric submanifolds satisfying Chen's equality in a real space form.