## Journal of the Mathematical Society of Japan

### Hypersurfaces with isotropic Blaschke tensor

#### Abstract

Let Mm be an m-dimensional submanifold without umbilical points in the m+1-dimensional unit sphere Sm+1. Three basic invariants of Mm under the Möbius transformation group of Sm+1 are a 1-form Φ called Möbius form, a symmetric (0,2) tensor A called Blaschke tensor and a positive definite (0,2) tensor g called Möbius metric. We call the Blaschke tensor is isotropic if there exists a function λ such that A = λg. One of the basic questions in Möbius geometry is to classify the hypersurfaces with isotropic Blaschke tensor. When λ is constant, the classification was given by Changping Wang and others. When λ is not constant, all hypersurfaces with dimensional m ≥ 3 and isotropic Blaschke tensor are explicitly expressed in this paper. Therefore, for the dimensional m ≥ 3, the above basic question is completely answered.

#### Article information

Source
J. Math. Soc. Japan, Volume 63, Number 4 (2011), 1155-1186.

Dates
First available in Project Euclid: 27 October 2011

https://projecteuclid.org/euclid.jmsj/1319721138

Digital Object Identifier
doi:10.2969/jmsj/06341155

Mathematical Reviews number (MathSciNet)
MR2855810

Zentralblatt MATH identifier
1242.53010

Subjects
Primary: 53A30: Conformal differential geometry

Keywords
Möbius geometry Blaschke tensor

#### Citation

GUO, Zhen; FANG, Jianbo; LIN, Limiao. Hypersurfaces with isotropic Blaschke tensor. J. Math. Soc. Japan 63 (2011), no. 4, 1155--1186. doi:10.2969/jmsj/06341155. https://projecteuclid.org/euclid.jmsj/1319721138

#### References

• M. A. Akivis and V. V. Goldberg, Conformal differential geometry and its generalizations, Wiley, New York, 1996.
• M. A. Akivis and V. V. Goldberg, A conformal differential invariant and the conformal rigidity of hypersurfaces, Proc. Amer. Math. Soc., 125 (1997), 2415–2424.
• W. Blaschke, Vorlesungen über Differentialgeometrie, 3, Springer-Verlag, Berlin, 1929.
• B. Y. Chen, Total mean curvature and submanifolds of finite type, World Scientific, Singapore, 1984.
• Z. Guo, H. Li and C. P. Wang, The Möbius characterizations of Willmore tori and Veronese submanifolds in the unit sphere, Pacific J. Math., 241 (2009), 227–242.
• Z. Guo, H. Li and C. P. Wang, The second variation formula for Willmore submanifolds in $S^{n}$, Results Math., 40 (2001), 205–225.
• Z. J. Hu and H. Li, Submanifolds with constant Möbius scalar curvature in $S^{n}$, Manuscripta Math., 111 (2003), 287–302.
• Z. J. Hu and H. Li, Classification of hypersurfaces with parallel Möbius second fundamental form in $S^{n+1}$, Sci. China Ser. A, 34 (2004), 28–39.
• H. Li, H. L. Liu, C. P. Wang and G. S. Zhao, Möbius isoparametric hypersurfaces in $S^{n+1}$ with two principal curvature, Acta Math. Sin. (Engl. Ser.), 18 (2002), 437–446.
• H. Li and C. P. Wang, Möbius geometry of hypersurfaces with constant mean curvature and scalar curvature, Manuscripta Math., 112 (2003), 1–13.
• H. Li and C. P. Wang, Surfaces with vanishing Möbius form in $S^n$, Acta Math. Sin. (Engl. Ser.), 19 (2003), 671–678.
• H. Li, C. P. Wang and F. E. Wu, A Möbius characterization of Veronese surfaces in $S^n$, Math. Ann., 319 (2001), 707–714.
• H. L. Liu, C. P. Wang and G. S. Zhao, Möbius isotropic submanifolds in $S^{n}$, Tohoku Math. J., 53 (2001), 553–569.
• F. J. Pedit and T. J. Willmore, Conformal Geometry, Atti Sem. Mat. Fis. Univ. Modena, XXXVI (1988), 237–245.
• C. P. Wang, Möbius geometry of submanifolds in $S^{n}$, Manuscripta Math., 96 (1998), 517–534.
• T. J. Willmore, Total curvature in Riemannian geometry, Ellis Horwood, Chichester, 1982.