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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



Submanifolds and a pinching problem on the second fundamental tensors

Author: Masafumi Okumura
Journal: Trans. Amer. Math. Soc. 178 (1973), 285-291
MSC: Primary 53C40
MathSciNet review: 0317246
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Abstract: This paper gives a sufficient condition for a submanifold of a Riemannian manifold of nonnegative constant curvature to be totally umbilical. The condition will be given by an inequality which is established between the length of the second fundamental tensor and the mean curvature.

References [Enhancements On Off] (What's this?)

    J. A. Erbacher, Isometric immersions of Riemannian manifolds into space forms, Thesis, Brown University, Providence, R. I., 1970. H. Liebmann, Über die Verbiegung der geschlassenen Flächen positiver Krummung, Math. Ann. 53 (1900), 91-112.
  • Katsumi Nomizu and Brian Smyth, A formula of Simons’ type and hypersurfaces with constant mean curvature, J. Differential Geometry 3 (1969), 367–377. MR 266109
  • Masafumi Okumura, Hypersurfaces and a pinching problem on the second fundamental tensor, Amer. J. Math. 96 (1974), 207–213. MR 353216, DOI
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Keywords: Second fundamental tensor, totally umbilical submanifold, pinching
Article copyright: © Copyright 1973 American Mathematical Society