Generalized Ejiri's Rigidity Theorem for Submanifolds in Pinched Manifolds |
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Authors: | Hongwei XU Li LEI and Juanru GU |
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Institution: | Center of Mathematical Sciences, Zhejiang University, Hangzhou 310027, China.,Center of Mathematical Sciences, Zhejiang University, Hangzhou 310027, China. and Corresponding author. Department of Applied Mathematics, Zhejiang University of Technology, Hangzhou 310023, China; Center of Mathematical Sciences, Zhejiang University, Hangzhou 310027, China. |
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Abstract: | Let $M^{n}(n\geq4)$ 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
$\delta(n,p)\in(0,1)$ such that if the sectional curvature of $N$
satisfies $\ov{K}_{N}\in\delta(n,p), 1]$, 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 $S^n\big(\frac{1}{\sqrt{1+H^2}}\big)$, a
Clifford hypersurface
$S^{m}\big(\frac{1}{\sqrt{2(1+H^2)}}\big)\times
S^{m}\big(\frac{1}{\sqrt{2(1+H^2)}}\big)$ in the totally umbilic
sphere $S^{n+1}\big(\frac{1}{\sqrt{1+H^2}}\big)$ with $n=2m$, or
$\mathbb{C}P^{2}\big(\frac{4}{3}(1+H^2)\big)$ in
$S^7\big(\frac{1}{\sqrt{1+H^2}}\big)$. This is a generalization of
Ejiri''s rigidity theorem. |
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Keywords: | Minimal submanifold Ejiri rigidity theorem Ricci curvature Mean curvature |
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