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

TL;DR

This paper classifies submanifolds in the unit sphere with a parallel Blaschke tensor, focusing on those with exactly three distinct eigenvalues, and introduces new examples in this geometric setting.

Contribution

It provides a complete classification of Blaschke parallel submanifolds with three eigenvalues and introduces new examples of such submanifolds.

Findings

Classification of Blaschke parallel submanifolds with three eigenvalues

Introduction of new examples of these submanifolds

Advancement in understanding M"obius invariants in sphere geometry

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.