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Proceedings of the American Mathematical Society

Published by the American Mathematical Society, the Proceedings of the American Mathematical Society (PROC) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6826 (online) ISSN 0002-9939 (print)

The 2020 MCQ for Proceedings of the American Mathematical Society is 0.85.

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A sharp differentiable pinching theorem for submanifolds in space forms
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by Juan-Ru Gu and Hong-Wei Xu PDF
Proc. Amer. Math. Soc. 144 (2016), 337-346 Request permission

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 $n=4$.
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Additional 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