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Complete hypersurfaces with $ RS=0$ in $ E\sp{n+1}$


Author: Yoshio Matsuyama
Journal: Proc. Amer. Math. Soc. 88 (1983), 119-123
MSC: Primary 53B25; Secondary 53C40
DOI: https://doi.org/10.1090/S0002-9939-1983-0691290-9
MathSciNet review: 691290
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Abstract: A locally symmetric Riemannian manifold satisfies $ RR = 0$ and in particular $ RS = 0$. The purpose of this paper is to show that the conditions $ RR = 0$ and $ RS = 0$ are equivalent for complete hypersurfaces in $ {E^{n + 1}}$ and to give by $ RS = 0$ some characterizations of locally symmetric hypersurfaces in $ {E^{n + 1}}$.


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

  • [1] E. Cartan, Sur quelques familles remarquables d'hypersurfaces, C. R. Cong. Math. Liege (1939), 30-41; Oeuvres complètes, Tome III, Vol. 2, p. 1481.
  • [2] Y. Matsuyama, Hypersurfaces with $ RS = 0$ in a Euclidean space, Bull. Fac. Sci. Engrg. Chuo Univ. 24 (1981), 13-19. MR 655701 (84a:53005)
  • [3] R. Miyaoka, Complete hypersurfaces in the space form with three principal curvatures, Math. Z. 179 (1982), 345-354. MR 649037 (83k:53076)
  • [4] R. Naka-Miyaoka, Minimal hypersurfaces in the space form with three principal curvatures, Math. Z. 170 (1980), 137-151. MR 562583 (82m:53058)
  • [5] K. Nomizu, On hypersurfaces satisfying a certain condition on the curvature tensor, Tôhoku Math. J. 20 (1968), 46-59. MR 0226549 (37:2138)
  • [6] P. J. Ryan, Homogeneity and some curvature conditions for hypersurfaces, Tôhoku Math. J. 21 (1969), 363-388. MR 0253243 (40:6458)
  • [7] -, Hypersurfaces with parallel Ricci tensor, Osaka J. Math. 8 (1971), 251-259. MR 0296859 (45:5918)
  • [8] S. Tanno, Hypersurfaces satisfying a certain condition on the Ricci tensor, Tôhoku Math. J. 21 (1969), 297-303. MR 0261508 (41:6121)

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Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1983-0691290-9
Keywords: Hypersurfaces, curvature tensor, Ricci tensor, complete
Article copyright: © Copyright 1983 American Mathematical Society

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