A classification of immersed hypersurfaces in spheres with parallel Blaschke tensor

Abstract

As is known, the Blaschke tensor A (a symmetric covariant 2-tensor) is one of the fundamental Möbius invariants in the Möbius differential geometry of submanifolds in the unit sphere S n , and the eigenvalues of A are referred to as the Blaschke eigenvalues. In this paper, we shall prove a classification theorem for immersed umbilic-free submanifolds in S n with a parallel Blaschke tensor. For proving this classification, some new kinds of examples are first defined.

Möbius isoparametric hypersurfaces with three distinct principal curvatures

“…Then we easily see that the Blaschke tensor is also parallel. Now, according to [Li and Zhang 2006], we have˜ = 0.…”

“…In many interesting situations, we find that = 0 is a very natural condition. For details, we refer to [Guo et al 2001;Liu et al 2001;Li et al 2002;Li and Wang 2003a;Hu and Li 2003;2004;2005a;2005b;Hu et al 2007;Li and Zhang 2006;, a series of nice results established in recent years under the condition = 0.…”

Classification of the Blaschke Isoparametric Hypersurfaces with Three Distinct Blaschke Eigenvalues

“…In this direction, there have been some interesting results, for example, classifications of hypersurfaces and submanifolds with Blaschke tensors linearly dependent on the Möbius metrics and the second Möbius fundamental forms [14,15], generalizing the classification of Möbius isotropic submanifolds [16], and the classification of hypersurfaces with parallel Blaschke tensors [17]. As generalizations of these results, classification theorems concerning the so called "para-Blaschke tensor" D λ := A + λB with λ a constant have been obtained in [18,19].…”

On hypersurfaces with parallel Möbius form and constant para-Blaschke eigenvalues

“…Theorem 1.1 in [15], together with [12]). Similarly, after many partial results in [7,[17][18][19][20][21], Li and Wang [16] also proved that a Blaschke isoparametric hypersurface in S n+1 with more than two distinct Blaschke eigenvalues is Möbius isoparametric.…”