Complete hypersurfaces with $RS=0$ in $E^{n+1}$
HTML articles powered by AMS MathViewer
- by Yoshio Matsuyama PDF
- Proc. Amer. Math. Soc. 88 (1983), 119-123 Request permission
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
-
E. Cartan, Sur quelques familles remarquables d’hypersurfaces, C. R. Cong. Math. Liege (1939), 30-41; Oeuvres complètes, Tome III, Vol. 2, p. 1481.
- Yoshio Matsuyama, Hypersurfaces with $RS=0$ in a Euclidean space, Bull. Fac. Sci. Engrg. Chuo Univ. 24 (1981), 13–19. MR 655701
- Reiko Miyaoka, Complete hypersurfaces in the space form with three principal curvatures, Math. Z. 179 (1982), no. 3, 345–354. MR 649037, DOI 10.1007/BF01215336
- Reiko Naka-Miyaoka, Minimal hypersurfaces in the space form with three principal curvatures, Math. Z. 170 (1980), no. 2, 137–151. MR 562583, DOI 10.1007/BF01214769
- Katsumi Nomizu, On hypersurfaces satisfying a certain condition on the curvature tensor, Tohoku Math. J. (2) 20 (1968), 46–59. MR 226549, DOI 10.2748/tmj/1178243217
- Patrick J. Ryan, Homogeneity and some curvature conditions for hypersurfaces, Tohoku Math. J. (2) 21 (1969), 363–388. MR 253243, DOI 10.2748/tmj/1178242949
- Patrick J. Ryan, Hypersurfaces with parallel Ricci tensor, Osaka Math. J. 8 (1971), 251–259. MR 296859
- Shûkichi Tanno, Hypersurfaces satisfying a certain condition on the Ricci tensor, Tohoku Math. J. (2) 21 (1969), 297–303. MR 261508, DOI 10.2748/tmj/1178242998
Additional Information
- © Copyright 1983 American Mathematical Society
- 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