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

TL;DR

This paper classifies immersed umbilic-free submanifolds in the unit sphere with a parallel Blaschke tensor, introducing new examples and advancing the understanding of M"obius invariants in differential geometry.

Contribution

It provides a classification theorem for such submanifolds and introduces new examples, enriching the study of Blaschke tensors in M"obius geometry.

Findings

Classification theorem for submanifolds with parallel Blaschke tensor

Introduction of new examples of such submanifolds

Deeper understanding of M"obius invariants in differential 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 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.