Generalized Ejiri’s Rigidity Theorem for Submanifolds in Pinched Manifolds

Abstract

Let M ⁿ(n ≥ 4) be an oriented compact submanifold with parallel mean curvature in an (n + p)-dimensional complete simply connected Riemannian manifold N ^n+p. Then there exists a constant δ(n, p) 2 (0, 1) such that if the sectional curvature of N satisfies ${\overline K _N} \in \;\,\left[ {\delta \left( {n,p} \right),\;1} \right]$, and if M has a lower bound for Ricci curvature and an upper bound for scalar curvature, then N is isometric to S ^n+p. Moreover, M is either a totally umbilic sphere ${S^n}({1 \over {\sqrt {1 + {H^2}} }})$, a Clifford hypersurface ${S^m}({1 \over {\sqrt {2(1 + {H^2})} }})\; \times \;{S^m}({1 \over {\sqrt {2(1 + {H^2})} }})$ in the totally umbilic sphere ${S^{n+1}}({1 \over {\sqrt {1 + {H^2}} }})$ with n = 2m, or ${\mathbb{C}{\rm{P}}^2}\left( {{4 \over 3}\left( {1 + {H^2}} \right)} \right)$ in ${S^7}({1 \over {\sqrt {1 + {H^2}} }})$. This is a generalization of Ejiri’s rigidity theorem.

Keywords

Minimal submanifold / Ejiri rigidity theorem / Ricci curvature / Mean curvature

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