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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

The differentiable sphere theorem for manifolds with positive Ricci curvature


Authors: Hong-Wei Xu and Juan-Ru Gu
Journal: Proc. Amer. Math. Soc. 140 (2012), 1011-1021
MSC (2010): Primary 53C20; Secondary 53C40
Published electronically: July 21, 2011
MathSciNet review: 2869085
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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$.


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Additional Information

Hong-Wei Xu
Affiliation: Center of Mathematical Sciences, Zhejiang University, Hangzhou 310027, People’s Republic of China
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

DOI: http://dx.doi.org/10.1090/S0002-9939-2011-10952-3
PII: S 0002-9939(2011)10952-3
Keywords: Compact manifolds, differentiable sphere theorem, Ricci curvature, Ricci flow, second fundamental form.
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
Article copyright: © Copyright 2011 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.