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Transactions of the American Mathematical Society

Published by the American Mathematical Society, the Transactions of the American Mathematical Society (TRAN) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

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

The 2020 MCQ for Transactions of the American Mathematical Society is 1.43.

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Submanifolds and a pinching problem on the second fundamental tensors
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by Masafumi Okumura PDF
Trans. Amer. Math. Soc. 178 (1973), 285-291 Request permission

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
    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 10.2307/2373587
  • K. Yano and S. Bochner, Curvature and Betti numbers, Annals of Mathematics Studies, No. 32, Princeton University Press, Princeton, N. J., 1953. MR 0062505
  • Kentaro Yano and Shigeru Ishihara, Submanifolds with parallel mean curvature vector, J. Differential Geometry 6 (1971/72), 95–118. MR 298598
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Additional Information
  • © Copyright 1973 American Mathematical Society
  • Journal: Trans. Amer. Math. Soc. 178 (1973), 285-291
  • MSC: Primary 53C40
  • DOI: https://doi.org/10.1090/S0002-9947-1973-0317246-1
  • MathSciNet review: 0317246