Hypersurfaces with isotropic para-Blaschke tensor

Abstract

Let Mn be an n-dimensional submanifold without umbilical points in the (n + 1)-dimensional unit sphere Sn+1. Four basic invariants of Mn under the Moebius transformation group of Sn+1 are a 1-form Φ called moebius form, a symmetric (0, 2) tensor A called Blaschke tensor, a symmetric (0, 2) tensor B called Moebius second fundamental form and a positive definite (0, 2) tensor g called Moebius metric. A symmetric (0, 2) tensor D = A + µB called para-Blaschke tensor, where µ is constant, is also an Moebius invariant. We call the para-Blaschke tensor is isotropic if there exists a function λ such that D = λg. One of the basic questions in Moebius geometry is to classify the hypersurfaces with isotropic para-Blaschke tensor. When λ is not constant, all hypersurfaces with isotropic para-Blaschke tensor are explicitly expressed in this paper.