| Title:
|
Three-dimensional curvature homogeneous hypersurfaces (English) |
| Author:
|
Calvaruso, G. |
| Author:
|
Marinosci, R. A. |
| Author:
|
Perrone, D. |
| Language:
|
English |
| Journal:
|
Archivum Mathematicum |
| ISSN:
|
0044-8753 (print) |
| ISSN:
|
1212-5059 (online) |
| Volume:
|
36 |
| Issue:
|
4 |
| Year:
|
2000 |
| Pages:
|
269-278 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| 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. (English) |
| Keyword:
|
curvature homogeneous spaces |
| Keyword:
|
isoparametric hypersurfaces |
| MSC:
|
53C20 |
| MSC:
|
53C30 |
| MSC:
|
53C40 |
| idZBL:
|
Zbl 1054.53070 |
| idMR:
|
MR1811171 |
| . |
| Date available:
|
2008-06-06T22:26:14Z |
| Last updated:
|
2012-05-10 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/107741 |
| . |
| Reference:
|
[1] Boeckx E., Kowalski O., Vanhecke L.,: Riemannian manifolds of conullity two.World Scientific 1996. Zbl 0904.53006, MR 1462887 |
| Reference:
|
[2] Calvaruso G., Vanhecke L.: Special ball-homogeneous spaces.Z. Anal. Anwendungen (4) 16 (1997), 789-800. Zbl 0892.53023, MR 1615680 |
| Reference:
|
[3] Calvaruso G., Vanhecke L.: Ball-homogeneous spaces.Proceedings of the Workshop on Differential Geometry, Santiago 89 (1998), 35-51. Zbl 0912.53034 |
| Reference:
|
[4] Cartan E.: Sur des familles remarquables d’hypersurfaces isoparamétriques dans les espaces sphériques.Math. Z. 45 (1939), 335-367. Zbl 0021.15603, MR 0000169 |
| Reference:
|
[5] Cecil T. E., Ryan P. J.: Tight and taut immersions of manifolds.Research Notes in Math., Pitman 1985. Zbl 0596.53002, MR 0781126 |
| Reference:
|
[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 |
| Reference:
|
[7] Ferus D.: Notes on isoparametric hypersurfaces.Escola de Geometria Diferencial, Universidade Estadual de Campinas, 1980. |
| Reference:
|
[8] Ferus D., Karcher H., Münzner H. F.: Clifford algebren und neue isoparametrische hyperflächen.Math. Z. (1981), 479-502. |
| Reference:
|
[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 |
| Reference:
|
[10] Ozeki H., Takeuchi M.: On some types of isoparametric hypersurfaces in spheres, I.Tôhoku Math. J. 27 (1975), 515-559. Zbl 0359.53011, MR 0454888 |
| Reference:
|
[11] Ozeki H., Takeuchi M.: On some types of isoparametric hypersurfaces in spheres, I.Tôhoku Math. J. 28 (1976), 7-55. MR 0454889 |
| Reference:
|
[12] Sekigawa K.: On some $3$-dimensional Riemannian manifolds.Hokkaido Math. J. 26 (1974), 259-270. MR 0353204 |
| Reference:
|
[13] Singer I. M.: Infinitesimally homogeneous spaces.Comm. Pure Appl. Math. 13 (1960), 685-697. Zbl 0171.42503, MR 0131248 |
| Reference:
|
[14] Takagi H.: On curvature homogeneity of Riemannian manifolds.Tôhoku Math. J. 26 (1974), 581-585. Zbl 0302.53022, MR 0365417 |
| Reference:
|
[15] Tricerri F., Vanhecke L.: Cartan hypersurfaces and reflections.Nihonkai Math. J. (2) 1 (1990), 203-208. Zbl 0956.53505, MR 1090781 |
| Reference:
|
[16] Tsukada K.: Curvature homogeneous hypersurfaces immersed in a real space form.Tôhoku Math. J. 40 (1988), 221-244. Zbl 0651.53037, MR 0943821 |
| Reference:
|
[17] Yamada A.: Homogeneity of hypersurfaces in a sphere.Tsukuba J. Math. 22 (1) (1998), 131-143. Zbl 0981.53040, MR 1637672 |
| Reference:
|
[18] Yamato K.: A characterization of locally homogeneous Riemann manifolds of dimension $3$.Nagoya Math. J. 123 (1991), 77-90. Zbl 0738.53032, MR 1126183 |
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