A sharp differentiable pinching theorem for submanifolds in space forms
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- by Juan-Ru Gu and Hong-Wei Xu
- Proc. Amer. Math. Soc. 144 (2016), 337-346
- DOI: https://doi.org/10.1090/proc/12908
- Published electronically: September 11, 2015
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Abstract:
Let $M$ be an $n$-dimensional compact submanifold in the simply connected space form $F^{n+p}(c)$ with $c+H^2>0$. We verify that if the sectional curvature of $M$ satisfies $K_{M} > \frac {n-2}{n+2}c+\frac {n^2H^2}{8(n+2)},$ then $M$ is diffeomorphic to a spherical space form. Moreover, we show that if $M$ is an oriented compact submanifold in $F^{n+p}(c)$ with $c\ge 0$, and if $n\neq 3,5$, $K_{M} > \frac {n-2}{n+2}c+\frac {n^2H^2}{8(n+2)},$ then $M$ is diffeomorphic to the standard $n$-sphere. It should be emphasized that our results are optimal for $n=4$.References
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Bibliographic Information
- Juan-Ru Gu
- Affiliation: Center of Mathematical Sciences, Zhejiang University, Hangzhou, 310027, People’s Republic of China
- Email: gujr@cms.zju.edu.cn
- 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
- Received by editor(s): November 1, 2014
- Published electronically: September 11, 2015
- Additional Notes: This research was supported by the NSFC, Grant Nos. 11371315, 11301476, and 1153012; the trans5-CENTURY0 training Programme Foundation for Talents by the Ministry of Education of China; and the China Postdoctoral Science Foundation, Grant No. 2013T60582.
- Communicated by: Lei Ni
- © Copyright 2015 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 144 (2016), 337-346
- MSC (2010): Primary 53C20, 53C24, 53C40
- DOI: https://doi.org/10.1090/proc/12908
- MathSciNet review: 3415600