Journal of the Mathematical Society of Japan

Hypersurfaces with isotropic Blaschke tensor

Zhen GUO, Jianbo FANG, and Limiao LIN

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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

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

First available in Project Euclid: 27 October 2011

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 53A30: Conformal differential geometry
Secondary: 53B25: Local submanifolds [See also 53C40]

Möbius geometry Blaschke tensor


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.

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