Springer Nature is making SARS-CoV-2 and COVID-19 research free. View research | View latest news | Sign up for updates

Möbius homogeneous hypersurfaces with three distinct principal curvatures in Sn+1

  • 16 Accesses

  • 2 Citations

Abstract

Let x: M n → Sn+1 be an immersed hypersurface in the (n + 1)-dimensional sphere Sn+1. If, for any points p, qM n, there exists a Möbius transformation ϕ: Sn+1 → Sn+1 such that ϕx(M n) = x(M n) and ϕx(p) = x(q), then the hypersurface is called a Möbius homogeneous hypersurface. In this paper, the Möbius homogeneous hypersurfaces with three distinct principal curvatures are classified completely up to a Möbius transformation.

This is a preview of subscription content, log in to check access.

References

  1. [1]

    Akivis, M. A. and Goldberg, V. V., A conformal differential invariants and the conformal rigidity of hypersurfaces, Proc. Amer. Math. Soc., 125, 1997, 2415–2424.

  2. [2]

    Cartan, E., Sur des familes remarquables d’hypersurfaces isoparamétriques dans les espace sphériques, Math. Z., 45, 1939, 335–367.

  3. [3]

    Cecil, T. E., Lie Sphere Geometry: With Applications to Submanifolds, Springer-Verlag, New York, 1992.

  4. [4]

    Guo, Z., Li, H. and Wang, C. P., The Möbius characterizations of Willmore tori and Veronese submanifolds in unit sphere, Pacific J. Math., 241, 2009, 227–242.

  5. [5]

    Hu, Z. J. and Zhai, S. J., Möbius isoparametric hypersurfaces with three distinct principal curvatures,II, Pacific J. Math., 249, 2011, 343–370.

  6. [6]

    Li, T. Z., Ma, X. and Wang, C. P., Möbius homogeneous hypersurfaces with two distinct principal curvatures in Sn+1, Ark. Mat., 51, 2013, 315–328.

  7. [7]

    Li, X. X. and Zhang, F. Y., On the Blaschke isoparametric hypersurfaces in the unit sphere, Acta Math. Sin. (Engl. Ser.), 25, 2009, 657–678.

  8. [8]

    Liu, H., Wang, C. P. and Zhao, G. S., Möbius isotropic submanifolds in Sn, Tohoku Math. J., 53, 2001, 553–569.

  9. [9]

    O’Neil, B., Semi-Riemannian Geometry, Academic Press, New York, 1983.

  10. [10]

    Sulanke, R., Möbius geometry V: Homogeneous surfaces in the Möbius space S 3, Topics in Differential Geometry, Vol. I, II, Debrecen, 1984, 1141–1154, Colloq. Math. Soc. János Bolyai, 46, North-Holland, Amsterdam, 1988.

  11. [11]

    Wang, C. P., Möbius geometry of submanifolds in Sn, Manuscripta Math., 96, 1998, 517–534.

  12. [12]

    Wang, C. P., Möbius geometry for hypersurfaces in S4, Nagoya Math. J., 139, 1995, 1–20.

Download references

Acknowledgments

The author would like to thank Professor Jie Qing for his hospitality and help. The author would also like to thank the referee for some valuable suggestions.

Author information

Correspondence to Tongzhu Li.

Additional information

This work was supported by the National Natural Science Foundation of China (Nos. 11571037, 11471021).

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Li, T. Möbius homogeneous hypersurfaces with three distinct principal curvatures in Sn+1 . Chin. Ann. Math. Ser. B 38, 1131–1144 (2017). https://doi.org/10.1007/s11401-017-1028-2

Download citation

Keywords

  • Möbius transformation group
  • Conformal transformation group
  • Möbius homogeneous hypersurfaces
  • Möbius isoparametric hypersurfaces

2000 MR Subject Classification

  • 53A30
  • 53C40