Remote Access Proceedings of the American Mathematical Society
Green Open Access

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
DOI: https://doi.org/10.1090/S0002-9939-97-03707-6
MathSciNet review: 1363427
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

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 $S^{n-1}\times \mathbf {R}^{1}$.


References [Enhancements On Off] (What's this?)

  • 1. K. R. Cai, Topology of certain closed submanifolds in a Euclidean space, Chinese Ann. of Math. A8 (1987), 234- 241. MR 89g:53091
  • 2. Q. M. Cheng, H.Nakagawa, Totally umbilic hypersurfaces, Hiroshima Math. Jour. 20 (1990), 1-10. MR 91f:53054
  • 3. S. Y. Cheng, S. T. Yau, Differential equations on Riemannian manifolds and their geometric applications, Comm. on Pure and Appl. Math. 28 (1975), 333-354. MR 52:6608
  • 4. J. S. Kim, S. O. Kim, An estimate of sectional curvatures of hypersurfaces with positive Ricci curvatures, Proc. Edinburgh Math. Soc. 38 (series II), part I (1995), 167-170. MR 95m:53079
  • 5. S. Kobayashi, K.Nomizu, Foundations of Differential Geometry, 1969. MR 38:6501

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (1991): 53A07, 53C20

Retrieve articles in all journals with MSC (1991): 53A07, 53C20


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: https://doi.org/10.1090/S0002-9939-97-03707-6
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

American Mathematical Society