RICCI CURVATURE OF SUBMANIFOLDS OF AN S-SPACE FORM

Abstract

Involving the Ricci curvature and the squared mean curvature, we obtain a basic inequality for a submanifold of an S-space form tangent to structure vector fields. Equality cases are also discussed. As applications we find corresponding results for almost semi-invariant submanifolds, <TEX>$\theta$</TEX>-slant submanifolds, anti-invariant submanifold and invariant submanifolds. A necessary and sufficient condition for a totally umbilical invariant submanifold of an S-space form to be Einstein is obtained. The inequalities for scalar curvature and a Riemannian invariant <TEX>$\Theta_k$</TEX> of different kind of submanifolds of a S-space form <TEX>$\tilde{M}(c)$</TEX> are obtained.