A characterization of round spheres

Author:
Sung-Eun Koh

Journal:
Proc. Amer. Math. Soc. **126** (1998), 3657-3660

MSC (1991):
Primary 53C40, 53C42

DOI:
https://doi.org/10.1090/S0002-9939-98-04589-4

MathSciNet review:
1469418

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Abstract | References | Similar Articles | Additional Information

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 dimensional compact oriented manifold without boundary into dimensional Euclidean space, hyperbolic space or the open half sphere is a totally umbilic immersion if, for some the -th mean curvature does not vanish and the ratio is constant.

**1.**K. Amur,*On a characterization of the 2-sphere*, American Mathematical Monthly**78**(1971), 382-384. MR**43:5463****2.**E. F. Beckenbach, R. Bellman,*Inequalities*, Springer Verlag, Berlin, 1971. MR**33:236**(earlier ed.)**3.**I. Bivens,*Integral formulas and hyperspheres in a simply connected space form*, Proc. Amer. Math. Soc.**88**(1983), 113-118. MR**84k:53052****4.**W. Y. Hsiang, Z. H. Teng, W.C. Yu,*New examples of constant mean curvature immersions of (2k-1)-spheres into Euclidean 2k-space*, Ann. of Math.**117**(1983), 609-625. MR**84i:53057****5.**S. Montiel, A. Ros,*Compact hypersurfaces: The Alexandrov theorem for higher order mean curvatures*, Differential Geometry (B. Lawson, ed.), Pitman Mono. 52, Longman, New York, 1991, pp. 279-296. MR**93h:53062****6.**H. C. Wente,*Counterexample to a conjecture of H. Hopf*, Pacific J. Math.**121**(1986), 193-243. MR**87d:53013**

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Additional Information

**Sung-Eun Koh**

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

Email:
sekoh@kkucc.konkuk.ac.kr

DOI:
https://doi.org/10.1090/S0002-9939-98-04589-4

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