Isoparametric hypersurfaces in complex hyperbolic spaces. (English) Zbl 1372.53056
An isoparametric hypersurface of a Riemannian manifold is a hypersurface such that all its sufficiently close parallel hypersurfaces have constant mean curvature.
In [Torino Atti 54, 974–979 (1918; JFM 47.0700.05)], C. Somigliana studied isoparametric surfaces of the 3-dimensional Euclidean space in relation to a problem of geometric optics.
In [Atti Accad. Naz. Lincei, Rend., VI. Ser. 27, 203–207 (1938; Zbl 0019.18403)], B. Segre generalized this study classifying isoparametric hypersurfaces in any Euclidian space.
E. Cartan studied this problem in real space forms where he derived a classification in real hyperbolic spaces [Ann. Mat. Pura Appl. (4) 17, 177–191 (1938; Zbl 0020.06505)] and in spheres [Math. Z. 45, 335–367 (1939; Zbl 0021.15603)].
In the paper under review, the authors continue the study started by the authors mentioned above and give a classification of isoparametric hypersurfaces in complex hyperbolic spaces.
In [Torino Atti 54, 974–979 (1918; JFM 47.0700.05)], C. Somigliana studied isoparametric surfaces of the 3-dimensional Euclidean space in relation to a problem of geometric optics.
In [Atti Accad. Naz. Lincei, Rend., VI. Ser. 27, 203–207 (1938; Zbl 0019.18403)], B. Segre generalized this study classifying isoparametric hypersurfaces in any Euclidian space.
E. Cartan studied this problem in real space forms where he derived a classification in real hyperbolic spaces [Ann. Mat. Pura Appl. (4) 17, 177–191 (1938; Zbl 0020.06505)] and in spheres [Math. Z. 45, 335–367 (1939; Zbl 0021.15603)].
In the paper under review, the authors continue the study started by the authors mentioned above and give a classification of isoparametric hypersurfaces in complex hyperbolic spaces.
Reviewer: Andreea Olteanu (Bucureşti)
MSC:
| 53C40 | Global submanifolds |
| 53B25 | Local submanifolds |
| 53C12 | Foliations (differential geometric aspects) |
| 53C24 | Rigidity results |
References:
| [1] | Berndt, J., Real hypersurfaces with constant principal curvatures in complex hyperbolic space, J. Reine Angew. Math., 395, 132-141 (1989) · Zbl 0655.53046 |
| [2] | Berndt, J., Homogeneous hypersurfaces in hyperbolic spaces, Math. Z., 229, 589-600 (1998) · Zbl 0929.53025 |
| [3] | Berndt, J.; Brück, M., Cohomogeneity one actions on hyperbolic spaces, J. Reine Angew. Math., 541, 209-235 (2001) · Zbl 1014.53042 |
| [4] | Berndt, J.; Díaz-Ramos, J. C., Real hypersurfaces with constant principal curvatures in complex hyperbolic spaces, J. Lond. Math. Soc., 74, 778-798 (2006) · Zbl 1109.53057 |
| [5] | Berndt, J.; Díaz-Ramos, J. C., Real hypersurfaces with constant principal curvatures in the complex hyperbolic plane, Proc. Amer. Math. Soc., 135, 3349-3357 (2007) · Zbl 1129.53031 |
| [6] | Berndt, J.; Díaz-Ramos, J. C., Homogeneous hypersurfaces in complex hyperbolic spaces, Geom. Dedicata, 138, 129-150 (2009) · Zbl 1169.53041 |
| [7] | Berndt, J.; Tamaru, H., Cohomogeneity one actions on noncompact symmetric spaces of rank one, Trans. Amer. Math. Soc., 359, 3425-3438 (2007) · Zbl 1117.53041 |
| [8] | Berndt, J.; Console, S.; Olmos, C., Submanifolds and Holonomy, Chapman & Hall/CRC Res. Notes Math., vol. 434 (2003), Chapman & Hall/CRC: Chapman & Hall/CRC Boca Raton, FL · Zbl 1043.53044 |
| [9] | Berndt, J.; Tricerri, F.; Vanhecke, L., Generalized Heisenberg Groups and Damek-Ricci Harmonic Spaces, Lecture Notes in Math., vol. 1598 (1995), Springer-Verlag: Springer-Verlag Berlin · Zbl 0818.53067 |
| [10] | Burth, T., Isoparametrische Hyperflächen in Lorentz-Raumformen (1993), University of Cologne, Master Thesis |
| [11] | Cartan, É., Familles de surfaces isoparamétriques dans les espaces à courbure constante, Ann. Mat. Pura Appl. (4), 17, 177-191 (1938) · JFM 64.1361.02 |
| [12] | Cartan, É., Sur des familles remarquables d’hypersurfaces isoparamétriques dans les espaces sphériques, Math. Z., 45, 335-367 (1939) · JFM 65.0792.01 |
| [13] | Cecil, T. E., Isoparametric and Dupin hypersurfaces, SIGMA Symmetry Integrability Geom. Methods Appl., 4, Article 062 pp. (2008) · Zbl 1165.53354 |
| [14] | Cecil, T.; Chi, Q.-S.; Jensen, G., Isoparametric hypersurfaces with four principal curvatures, Ann. of Math. (2), 166, 1-76 (2007) · Zbl 1143.53058 |
| [15] | Chi, Q. S., Isoparametric hypersurfaces with four principal curvatures III, J. Differential Geom., 94, 469-504 (2013) · Zbl 1280.53053 |
| [16] | Díaz-Ramos, J. C.; Domínguez-Vázquez, M., Inhomogeneous isoparametric hypersurfaces in complex hyperbolic spaces, Math. Z., 271, 1037-1042 (2012) · Zbl 1258.53052 |
| [17] | Díaz-Ramos, J. C.; Domínguez-Vázquez, M., Isoparametric hypersurfaces in Damek-Ricci spaces, Adv. Math., 239, 1-17 (2013) · Zbl 1293.53065 |
| [18] | Díaz-Ramos, J. C.; Domínguez-Vázquez, M.; Kollross, A., Polar actions on complex hyperbolic spaces, Math. Z. (2017) · Zbl 1383.53040 |
| [19] | Domínguez-Vázquez, M., Isoparametric foliations on complex projective spaces, Trans. Amer. Math. Soc., 368, 2, 1211-1249 (2016) · Zbl 1342.53042 |
| [20] | Dorfmeister, J.; Neher, E., Isoparametric hypersurfaces, case \(g = 6, m = 1\), Comm. Algebra, 13, 2299-2368 (1985) · Zbl 0578.53041 |
| [21] | Ferus, D.; Karcher, H.; Münzner, H. F., Cliffordalgebren und neue isoparametrische Hyperflächen, Math. Z., 177, 479-502 (1981) · Zbl 0443.53037 |
| [22] | Hahn, J., Isoparametric hypersurfaces in the pseudo-Riemannian space forms, Math. Z., 187, 195-208 (1984) · Zbl 0529.53046 |
| [23] | Immervoll, S., On the classification of isoparametric hypersurfaces with four principal curvatures in spheres, Ann. of Math. (2), 168, 1011-1024 (2008) · Zbl 1176.53057 |
| [24] | Kimura, M., Real hypersurfaces and complex submanifolds in complex projective space, Trans. Amer. Math. Soc., 296, 137-149 (1986) · Zbl 0597.53021 |
| [25] | Lohnherr, M.; Reckziegel, H., On ruled real hypersurfaces in complex space forms, Geom. Dedicata, 74, 267-286 (1999) · Zbl 0932.53018 |
| [26] | Magid, M. A., Lorentzian isoparametric hypersurfaces, Pacific J. Math., 118, 1, 165-197 (1985) · Zbl 0561.53057 |
| [27] | Miyaoka, R., Isoparametric hypersurfaces with \((g, m) = (6, 2)\), Ann. of Math. (2), 177, 53-110 (2013) · Zbl 1263.53049 |
| [28] | Montiel, S., Real hypersurfaces of a complex hyperbolic space, J. Math. Soc. Japan, 37, 515-535 (1985) · Zbl 0554.53021 |
| [29] | O’Neill, B., The fundamental equations of a submersion, Michigan Math. J., 13, 459-469 (1966) · Zbl 0145.18602 |
| [30] | O’Neill, B., Semi-Riemannian Geometry with Applications to Relativity (1983), Academic Press · Zbl 0531.53051 |
| [31] | Reckziegel, H., On the problem whether the image of a given differentiable map into a Riemannian manifold is contained in a submanifold with parallel second fundamental form, J. Reine Angew. Math., 325, 87-104 (1981) · Zbl 0449.53045 |
| [32] | Segre, B., Famiglie di ipersuperficie isoparametriche negli spazi euclidei ad un qualunque numero di dimensioni, Atti Accad. Naz. Lincei Rend. Cl. Sci. Fis. Mat. Natur. (6), 27, 203-207 (1938) · Zbl 0019.18403 |
| [33] | Siffert, A., Classification of isoparametric hypersurfaces in spheres with \((g, m) = (6, 1)\), Proc. Amer. Math. Soc., 144, 2217-2230 (2016) · Zbl 1343.53051 |
| [34] | Somigliana, C., Sulle relazioni fra il principio di Huygens e l’ottica geometrica, Atti Accad. Sci. Torino, LIV, 974-979 (1918-1919) · JFM 47.0700.05 |
| [35] | Stolz, S., Multiplicities of Dupin hypersurfaces, Invent. Math., 138, 253-279 (1999) · Zbl 0944.53035 |
| [36] | Takagi, R., Real hypersurfaces in a complex projective space with constant principal curvatures, J. Math. Soc. Japan, 27, 43-53 (1975) · Zbl 0292.53042 |
| [37] | Takagi, R., Real hypersurfaces in a complex projective space with constant principal curvatures, II, J. Math. Soc. Japan, 27, 507-516 (1975) · Zbl 0311.53064 |
| [38] | Thorbergsson, G., A survey on isoparametric hypersurfaces and their generalizations, (Handbook of Differential Geometry, vol. I (2000), North-Holland: North-Holland Amsterdam), 963-995 · Zbl 0979.53002 |
| [39] | Xiao, L., Lorentzian isoparametric hypersurfaces in \(H_1^{n + 1}\), Pacific J. Math., 189, 377-397 (1999) · Zbl 0923.53023 |
| [40] | Yau, S.-T., Open problems in geometry, (Greene, R.; Yau, S.-T., Differential Geometry: Partial Differential Equations on Manifolds. Differential Geometry: Partial Differential Equations on Manifolds, Los Angeles, 1990. Differential Geometry: Partial Differential Equations on Manifolds. Differential Geometry: Partial Differential Equations on Manifolds, Los Angeles, 1990, Proc. Sympos. Pure Math., vol. 54 (1993), Amer. Math. Soc.: Amer. Math. Soc. Providence, RI), Part 1 · Zbl 0984.53003 |
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