Abstract
Let x be an m-dimensional umbilic-free hypersurface in an (m + 1)-dimensional unit sphere \(\mathbb{S}^{m + 1}\) (m ⩾ 3). In this paper, we classify and explicitly express the hypersurfaces with two distinct principal curvatures and closed Möbius form, and then we characterize and classify conformally flat hypersurfaces of dimension larger than 3.
Similar content being viewed by others
References
Cartan E. La déformation des hypersurfaces dans l’éspace conforme réel à n ⩾ 5 dimensions. Bull Soc Math France, 1917, 45: 57–121
Akivis M A, Goldberg V V. Conformal differential geometry and its generalizations. New York: Wiley, 1996
Thorbergsson G. Dupin hypersurfaces. Bull London Math Soc, 1983, 15: 493–498
Cecil T E, Ryan P J. Tight and taut immersions of manifolds. In: Research Notes in Mathematics 107. London: Pitman Advanced Publishing Program, 1985
Pinkall U. Dupin hypersurfaces. Math Ann, 1985, 270: 427–440
Wang C P. Möbius geometry of submanifolds in \(\mathbb{S}^n\). Manuscripta Math, 1998, 96: 517–534
Akivis M A, Goldberg V V. A conformal differential invariant and the conformal rigedity of hypersurfaces. Proc Amer Math Soc, 1997, 125: 2415–2424
Hu Z J, Li H. Submanifolds with constant Möbius scalar curvature in S n. Manuscripta Math, 2003, 111: 287–302
Hu Z J, Li H. Classification of hypersurfaces with parallel Möbius second fundamental form in S n+1. Sci China Ser A, 2004, 34: 28–39
Liu H L, Wang C P, Zhao G S. Möbius isoparametric hypersurfaces in \(\mathbb{S}^n\). Tohoku Math J, 2001, 53: 553–569
Li H, Liu H L, Wang C P, et al. Möbius isoparametric hypersurfaces in S n+1 with two principal curvatures. Acta Math Sinica Engl Ser, 2002, 18: 437–446
Li H, Wang C P. Surfaces with vanishing Möbius form in S n. Acta Math Sinica Engl Ser, 2003, 19: 671–678
Guo Z, Li H, Wang C P. The Möbius Characterizations of Willmore tori and Veronese submanifolds in the unit sphere. Pacific J Math, 2009, 241: 227–242
Guo Z, Li H, Wang C P. The second variation formula for Willmore submanifolds in S n. Results Math, 2001, 40: 205–225
Guo Z, Fang J B, Lin L M. Hypersurfaces with isotropic Blaschke tensor. J Math Soc Japan, 2011, 63: 1155–1186
Guo Z, Li T Z, Lin L M, et al. The classification of the hypersurfaces with constant Möbius curvature. Math Z, doi:10.1007/s00209-011-0860-4
Verstraelen L, Zafindratrafa G. Some comments on Conformally flat submanifolds. Geometry and Topology, III: 131–147
Pedit F J, Willmore T J. Conformal Geometry. Atti Sem Mat Fis Univ Modena, 1988, XXXVI: 237–245
Chen B Y. Total Mean Curvature and Submanifolds of Finite Type. Singapore: World Scientific, 1984
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Lin, L., Guo, Z. Classification of hypersurfaces with two distinct principal curvatures and closed Möbius form in \(\mathbb{S}^{m + 1}\) . Sci. China Math. 55, 1463–1478 (2012). https://doi.org/10.1007/s11425-012-4391-1
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11425-012-4391-1