Previous |  Up |  Next

Article

Keywords:
curvature homogeneous spaces; isoparametric hypersurfaces
Summary:
This paper is motivated by the open problem whether a three-dimensional curvature homogeneous hypersurface of a real space form is locally homogeneous or not. We give some partial positive answers.
References:
[1] Boeckx E., Kowalski O., Vanhecke L.,: Riemannian manifolds of conullity two. World Scientific 1996. MR 1462887 | Zbl 0904.53006
[2] Calvaruso G., Vanhecke L.: Special ball-homogeneous spaces. Z. Anal. Anwendungen (4) 16 (1997), 789-800. MR 1615680 | Zbl 0892.53023
[3] Calvaruso G., Vanhecke L.: Ball-homogeneous spaces. Proceedings of the Workshop on Differential Geometry, Santiago 89 (1998), 35-51. Zbl 0912.53034
[4] Cartan E.: Sur des familles remarquables d’hypersurfaces isoparamétriques dans les espaces sphériques. Math. Z. 45 (1939), 335-367. MR 0000169 | Zbl 0021.15603
[5] Cecil T. E., Ryan P. J.: Tight and taut immersions of manifolds. Research Notes in Math., Pitman 1985. MR 0781126 | Zbl 0596.53002
[6] Chang S.: A closed hypersurface with constant scalar and mean curvature in $S^4$ is isoparametric. Comm. Anal. Geom. (1) 1 (1993), 71-100. MR 1230274
[7] Ferus D.: Notes on isoparametric hypersurfaces. Escola de Geometria Diferencial, Universidade Estadual de Campinas, 1980.
[8] Ferus D., Karcher H., Münzner H. F.: Clifford algebren und neue isoparametrische hyperflächen. Math. Z. (1981), 479-502.
[9] Kowalski O.: A classification of Riemannian $3$-manifolds with constant principal Ricci curvatures $\varrho _1 =\varrho _2 \ne \varrho _3$. Nagoya Math. J. 132 (1993), 1-36. MR 1253692
[10] Ozeki H., Takeuchi M.: On some types of isoparametric hypersurfaces in spheres, I. Tôhoku Math. J. 27 (1975), 515-559. MR 0454888 | Zbl 0359.53011
[11] Ozeki H., Takeuchi M.: On some types of isoparametric hypersurfaces in spheres, I. Tôhoku Math. J. 28 (1976), 7-55. MR 0454889
[12] Sekigawa K.: On some $3$-dimensional Riemannian manifolds. Hokkaido Math. J. 26 (1974), 259-270. MR 0353204
[13] Singer I. M.: Infinitesimally homogeneous spaces. Comm. Pure Appl. Math. 13 (1960), 685-697. MR 0131248 | Zbl 0171.42503
[14] Takagi H.: On curvature homogeneity of Riemannian manifolds. Tôhoku Math. J. 26 (1974), 581-585. MR 0365417 | Zbl 0302.53022
[15] Tricerri F., Vanhecke L.: Cartan hypersurfaces and reflections. Nihonkai Math. J. (2) 1 (1990), 203-208. MR 1090781 | Zbl 0956.53505
[16] Tsukada K.: Curvature homogeneous hypersurfaces immersed in a real space form. Tôhoku Math. J. 40 (1988), 221-244. MR 0943821 | Zbl 0651.53037
[17] Yamada A.: Homogeneity of hypersurfaces in a sphere. Tsukuba J. Math. 22 (1) (1998), 131-143. MR 1637672 | Zbl 0981.53040
[18] Yamato K.: A characterization of locally homogeneous Riemann manifolds of dimension $3$. Nagoya Math. J. 123 (1991), 77-90. MR 1126183 | Zbl 0738.53032
Partner of
EuDML logo