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| Generalized Ejiri's Rigidity Theorem for Submanifolds in Pinched Manifolds |
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Citation： |
Hongwei XU,Li LEI,Juanru GU.Generalized Ejiri's Rigidity Theorem for Submanifolds in Pinched Manifolds[J].Chinese Annals of Mathematics B,2020,41(2):285~302 |
| Page view： 902
Net amount： 741 |
Authors： |
Hongwei XU; Li LEI;Juanru GU |
Foundation： |
This work was supported by the National Natural Science
Foundation of China (Nos.11531012, 11371315, 11301476). |
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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. |
Keywords： |
Minimal submanifold, Ejiri rigidity theorem, Ricci curvature, Mean curvature |
Classification： |
53C40, 53C42 |
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