Proper Biconservative immersions into the Euclidean space

TL;DR

This paper classifies certain symmetric biconservative hypersurfaces in Euclidean space using equivariant differential geometry, showing that no proper biharmonic immersions exist within these classes.

Contribution

It provides a classification of invariant biconservative hypersurfaces and proves the non-existence of proper biharmonic immersions in these cases.

Findings

No proper biharmonic immersions in the studied invariant classes.

Classification of $SO(p+1) imes SO(q+1)$- and $SO(p+1)$-invariant biconservative hypersurfaces.

Use of equivariant differential geometry for classification.

Abstract

In this paper, using the framework of equivariant differential geometry, we study proper $SO(p+1) \times SO(q+1)$-invariant biconservative hypersurfaces into the Euclidean space ${\mathbb R}^n$ ($n=p+q+2$) and proper $SO(p+1)$-invariant biconservative hypersurfaces into the Euclidean space ${\mathbb R}^n$ ($n=p+2$). Moreover, we show that, in these two classes of invariant families, there exists no proper biharmonic immersion.