Ricci curvature of submanifolds of an $\mathcal{S}$-space form
Bull. Korean Math. Soc. 2009 Vol. 46, No. 5, 979-998
Printed September 1, 2009
Jeong-Sik Kim, Mohit Kumar Dwivedi, and Mukut Mani Tripathi
Yosu National University, Lucknow University, and Banaras Hindu University
Abstract : 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
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