Hostname: page-component-5c6d5d7d68-wpx84 Total loading time: 0 Render date: 2024-08-15T19:00:51.678Z Has data issue: false hasContentIssue false
  • English
  • Français

A classification of Riemannian 3-manifolds with constant principal ricci curvaturesρ1= ρ2≠ ρ3

Published online by Cambridge University Press:  22 January 2016

Oldřich Kowalski
Oldřich Kowalski*
Affiliation:
Faculty of Mathematics and Physics Charles UniversitySokolovská 83, 186 00 PrahaCzech Republic
Rights & Permissions [Opens in a new window]

Extract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

This paper has been motivated by various problems and results in differential geometry. The main motivation is the study of curvature homogeneous Riemannian spaces initiated in 1960 by I.M. Singer (see Section 9—Appendix for the precise definitions and references). Up to recently, only sporadic classes of examples have been known of curvature homogeneous spaces which are not locally homogeneous. For instance, isoparametric hypersurfaces in space forms give nice examples of nontrivial curvature homogeneous spaces (see [FKM]). To study the topography of curvature homogeneous spaces more systematically, it is natural to start with the dimension n = 3. The following results and problems have been particularly inspiring.

Type
Research Article
Information
Nagoya Mathematical Journal , Volume 132 , December 1993 , pp. 1 - 36
Copyright
Copyright © Editorial Board of Nagoya Mathematical Journal 1993

References

[B] Besse, A.L., Einstein manifolds, Springer-Verlag 1987.Google Scholar
[D] Turck, D. De., Existence of metrics with prescribed Ricci curvature: Local theory, Invent. Math., 65 (1981),179–201.Google Scholar
[F] Ferus, D., Karcher, H. and Münzner, H.F., Clifford-algebren und neue isoparametrische Hyperflächen, Math. Z., 177 (1981),479–502.CrossRefGoogle Scholar
[K1] Kowalski, O., An explicit classification of 3-dimensional Riemannian spaces satisfying R(X, Y)·R = 0, preprint, 1992.Google Scholar
[K2] Kowalski, O., A note to a theorem by K. Sekigawa, Comment. Math. Univ. Carolinae, 30, 1 (1989),85–88.Google Scholar
[KN1] Kobayashi, S. and Nomizu, K., Foundations of differential geometry Vol. I, Interscience Publishers, New York-London, 1963.Google Scholar
[KTV1] Kowalski, O., Tricerri, F. and Vanhecke, L., New examples of nonhomogeneous Riemannian manifolds whose curvature tensor is that of a Riemannian symmetric space, C. R. Acad. Sci. Paris, t. 311, Série I, (1990),355–360.Google Scholar
[KTV2] Kowalski, O., Tricerri, F. and Vanhecke, L., Curvature homogeneous Riemannian manifolds, J. Math. Pures Appl., 71 (1992),471–501.Google Scholar
[KTV3] Kowalski, O., Tricerri, F. and Vanhecke, L., Curvature homogeneous spaces with a solvable Lie group as homogeneous model, J. Math. Soc. Japan, 44, 3 (1992), 461–484.Google Scholar
[M] Milnor, J., Curvatures of left invariant metrics on Lie groups, Advances in Math., 21 (1976), 293–329.Google Scholar
[Se1] Sekigawa, K., On the Riemannian manifolds of the form B xfF , Kodai Math. Sem. Rep., 26 (1975), 343–347.Google Scholar
[Se2] Sekigawa, K., On some 3-dimensional curvature homogeneous spaces, Tensor, N.S. 31 (1977), 87–97.Google Scholar
[T] Takagi, H., On curvature homogeneity of Riemannian manifolds, Tôhoku Math. J., 26 (1974), 581–585.Google Scholar
[T] Tsukada, K., Curvature homogeneous hypersurfaces immersed in a real space form, Tôhoku Math. J., 40 (1988), 221–244.Google Scholar
[TV1] Tricerri, F. and Vanhecke, L., Curvature homogeneous Riemannian manifolds, Ann. Sci. École Norm. Sup., 22 (1989), 535–554.Google Scholar
[TV2] Tricerri, F. and Vanhecke, L., Approximating Riemannian Manifolds by Homogeneous Spaces, Proceedings of the Curvature Geometry Workshop, Lancaster 1989, pp.35–48.Google Scholar
[Y] Yamato, K., A characterization of locally homogeneous Riemann manifolds of dimension 3, Nagoya Math. J., 123 (1991), 77–90.Google Scholar