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

DOI:
https://doi.org/10.1090/S0002-9939-2011-10952-3

Published electronically:
July 21, 2011

MathSciNet review:
2869085

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Abstract: We prove that if is a compact Riemannian -manifold and if , where , , and are the sectional curvature and Ricci curvature of respectively, and , then is diffeomorphic to a spherical space form. In particular, if is a compact simply connected manifold with and , then is diffeomorphic to the standard -sphere . We also extend the differentiable sphere theorem above to submanifolds in Riemannian manifolds with codimension .

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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:
https://doi.org/10.1090/S0002-9939-2011-10952-3

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.