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

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 continue our job for the study on the submanifolds in $\bbs^n$ with parallel Blaschke tensors which we simply call {\em Blaschke parallel submanifolds} to find more examples and seek a complete classification finally. The main theorem of this paper is the classification of Blaschke parallel submanifolds in $\mathbb S^n$ with exactly three distinct Blaschke eigenvalues. Before proving this classification we define, as usual, a new class of examples.