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April, 2004 Spherical rigidities of submanifolds in Euclidean spaces
Qing-Ming CHENG
J. Math. Soc. Japan 56(2): 475-487 (April, 2004). DOI: 10.2969/jmsj/1191418640


In this paper, we study n-dimensional complete immersed submanifolds in a Euclidean space En+p. We prove that if Mn is an n-dimensional compact connected immersed submanifold with nonzero mean curvature H in En+p and satisfies either:

(1)Sn2H2n-1, or


then Mn is diffeomorphic to a standard n-sphere, where S and R denote the squared norm of the second fundamental form of Mn and the scalar curvature of Mn, respectively.

On the other hand, in the case of constant mean curvature, we generalized results of Klotz and Osserman [11] to arbitrary dimensions and codimensions; that is, we proved that the totally umbilical sphere Sn(c), the totally geodesic Euclidean space En, and the generalized cylinder Sn-1(c)×E1 are only n-dimensional (n>2) complete connected submanifolds Mn with constant mean curvature H in En+p if Sn2H2/(n-1) holds.


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Qing-Ming CHENG. "Spherical rigidities of submanifolds in Euclidean spaces." J. Math. Soc. Japan 56 (2) 475 - 487, April, 2004.


Published: April, 2004
First available in Project Euclid: 3 October 2007

zbMATH: 1066.53013
MathSciNet: MR2048469
Digital Object Identifier: 10.2969/jmsj/1191418640

Primary: 53C42

Keywords: differentiable sphere , generalized cylinder , locally convex hypersurfaces , mean curvature , squared norm of the second fundamental form , Submanifolds

Rights: Copyright © 2004 Mathematical Society of Japan


Vol.56 • No. 2 • April, 2004
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