Bulletin of the
Korean Mathematical Society BKMS

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

Keywords: $\mathcal{S}$-space form, almost semi-invariant submanifold, $\theta$-slant submanifold, anti-invariant submanifold, Ricci curvature, $k$-Ricci curvature, scalar curvature, $\Theta$-invaraint

MSC numbers: Primary 53C40, 53C15, 53C25