A characterization of round spheres

Koh, Sung-Eun

doi:10.1090/S0002-9939-98-04589-4

A characterization of round spheres
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Proc. Amer. Math. Soc. 126 (1998), 3657-3660 Request permission

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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