On the immersed submanifolds in the unit sphere with parallel Blaschke tensor

Abstract

As is known, the Blaschke tensor $A$ (a symmetric covariant $2$-tensor) is one of the fundamental M\"obius invariants in the M\"obius differential geometry of submanifolds in the unit sphere $\mathbb S^n$, 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 $\mathbb S^n$ with a parallel Blaschke tensor. For proving this classification, some new kinds of examples are first defined.