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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 structure of hypersurfaces
with some curvature conditions


Author: Ju Seon Kim
Journal: Proc. Amer. Math. Soc. 125 (1997), 1497-1501
MSC (1991): Primary 53A07; Secondary 53C20
MathSciNet review: 1363427
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Abstract: Let $M$ be a hypersurface in $\mathbf {R}^{n+1}$, and let $H, R$ denote the mean curvature and the scalar curvature of $M$ respectively. We show that if $M$ is compact and $R>\frac {n-2}{n-1}H^{2}$, then $M$ is diffeomorphic to $S^{n}$. Also we prove that if $M$ is complete, $H$ is constant and $R\geq \frac {n-2}{n-1}H^{2}$, then $M$ is $\mathbf {R}^{n}$ or $S^{n}$ or $S^{n-1}\times \mathbf {R}^{1}$.


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Additional Information

Ju Seon Kim
Affiliation: Department of Mathematics, Purdue University, West Lafayette, Indiana 47907
Address at time of publication: Department of Mathematics, Myong Ji University, Yongin, 449-728, Seoul, Korea

DOI: http://dx.doi.org/10.1090/S0002-9939-97-03707-6
PII: S 0002-9939(97)03707-6
Received by editor(s): May 17, 1995
Received by editor(s) in revised form: November 1, 1995
Communicated by: Christopher Croke
Article copyright: © Copyright 1997 American Mathematical Society