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Proceedings of the American Mathematical Society

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A characterization of round spheres

Author: Sung-Eun Koh
Journal: Proc. Amer. Math. Soc. 126 (1998), 3657-3660
MSC (1991): Primary 53C40, 53C42
MathSciNet review: 1469418
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Abstract: A new characterization of geodesic spheres in the simply connected space forms in terms of higher order mean curvatures is given: An immersion of an $n$ dimensional compact oriented manifold without boundary into $n+1$ dimensional Euclidean space, hyperbolic space or the open half sphere is a totally umbilic immersion if, for some $r,\ r=2,\dots ,n,$ the $(r-1)$-th mean curvature $H_{r-1}$ does not vanish and the ratio $H_{r}/H_{r-1}$ is constant.

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Sung-Eun Koh
Affiliation: Department of Mathematics, Kon-Kuk University, Seoul, 143-701, Korea

Keywords: Higher order mean curvature, principal curvature, umbilical point, Minkowski formula
Received by editor(s): April 25, 1997
Additional Notes: This research was supported by the KOSEF through Research Fund 96-0701-02-01-3, and by the Korean Ministry of Education through Research Fund BSRI-96-1438.
Communicated by: Christopher Croke
Article copyright: © Copyright 1998 American Mathematical Society