The differentiable sphere theorem for manifolds with positive Ricci curvature
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- by Hong-Wei Xu and Juan-Ru Gu PDF
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Abstract:
We prove that if $M^n$ is a compact Riemannian $n$-manifold and if $Ric_{\min }>(n-1)\tau _{n}K_{\max }$, where $K_{\max }(x):=\max _{\pi \subset T_{x}M}K(\pi )$, $Ric_{\min }(x):=\min _{u\in U_{x}M}Ric(u)$, $K(\cdot )$ and $Ric(\cdot )$ are the sectional curvature and Ricci curvature of $M$ respectively, and $\tau _{n}=1-\frac {6}{5(n-1)}$, then $M$ is diffeomorphic to a spherical space form. In particular, if $M$ is a compact simply connected manifold with $K\le 1$ and $Ric_M> (n-1)\tau _{n}$, then $M$ is diffeomorphic to the standard $n$-sphere $S^n$. We also extend the differentiable sphere theorem above to submanifolds in Riemannian manifolds with codimension $p$.References
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Additional Information
- Hong-Wei Xu
- Affiliation: Center of Mathematical Sciences, Zhejiang University, Hangzhou 310027, People’s Republic of China
- MR Author ID: 245171
- Email: xuhw@cms.zju.edu.cn
- Juan-Ru Gu
- Affiliation: Center of Mathematical Sciences, Zhejiang University, Hangzhou 310027, People’s Republic of China
- Email: gujr@cms.zju.edu.cn
- Received by editor(s): November 6, 2010
- Received by editor(s) in revised form: December 11, 2010
- Published electronically: July 21, 2011
- Additional Notes: Research supported by the NSFC, grant No. 10771187, 11071211, and the Trans-Century Training Programme Foundation for Talents by the Ministry of Education of China.
- Communicated by: Jianguo Cao
- © Copyright 2011
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - Journal: Proc. Amer. Math. Soc. 140 (2012), 1011-1021
- MSC (2010): Primary 53C20; Secondary 53C40
- DOI: https://doi.org/10.1090/S0002-9939-2011-10952-3
- MathSciNet review: 2869085