Journal of the Mathematical Society of Japan
- J. Math. Soc. Japan
- Volume 56, Number 2 (2004), 475-487.
Spherical rigidities of submanifolds in Euclidean spaces
In this paper, we study -dimensional complete immersed submanifolds in a Euclidean space . We prove that if is an -dimensional compact connected immersed submanifold with nonzero mean curvature in and satisfies either:
then is diffeomorphic to a standard -sphere, where and denote the squared norm of the second fundamental form of and the scalar curvature of , respectively.
On the other hand, in the case of constant mean curvature, we generalized results of Klotz and Osserman  to arbitrary dimensions and codimensions; that is, we proved that the totally umbilical sphere , the totally geodesic Euclidean space , and the generalized cylinder are only -dimensional complete connected submanifolds with constant mean curvature in if holds.
J. Math. Soc. Japan, Volume 56, Number 2 (2004), 475-487.
First available in Project Euclid: 3 October 2007
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CHENG, Qing-Ming. Spherical rigidities of submanifolds in Euclidean spaces. J. Math. Soc. Japan 56 (2004), no. 2, 475--487. doi:10.2969/jmsj/1191418640. https://projecteuclid.org/euclid.jmsj/1191418640