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

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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.

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  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.

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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.

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Correspondence to Tongzhu Li.

Additional information

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

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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

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

2000 MR Subject Classification

  • 53A30
  • 53C40