Extrinsic spheres in a real space form

Abstract

Let $M$ be an $n$-dimensional orientable compact hypersurface in an $(n+1)$-dimensional real space form $\overline{M}(c)$, $n\geq 2$. If the lengths $\left\| R\right\| $, $\left\| A\right\| $ and $\left\| \nabla \alpha \right\| $ of the curvature tensor field $R$, the shape operator $A$, the gradient $\nabla \alpha $ of the mean curvature $\alpha $ and the scalar curvature $S$ of the hypersurface $M$ satisfy the inequality \begin{equation*} \frac{1}{2}\left\| R\right\| ^{2}\leq cS+\delta \left\| A\right\| ^{2}-n(n-1)\left\| \nabla \alpha \right\| ^{2} \end{equation*} where $\delta =\min Ric=\underset{p\in M\quad v\in T_{p}M\quad \left\| v\right\| =1}{\min }Ric_{p}(v)$, $Ric$ is Ricci curvature of the hypersurface, then it is shown that $M$ is an extrinsic sphere in $\overline{M}(c)$. In particular we deduce that the condition $\frac{1}{2}\left\| R\right\| ^{2}\leq \delta \left\| A\right\| ^{2}-n(n-1)\left\| \nabla \alpha \right\| ^{2}$ characterizes spheres in the Euclidean space $R^{n+1}$ among the compact orientable hypersurfaces whose Ricci curvatures are bounded below by a constant $\delta >0$.