Biconservative ideal hypersurfaces in Euclidean spaces

TL;DR

This paper extends the classification of certain biconservative hypersurfaces in Euclidean spaces, showing that under specific conditions they must have constant mean curvature, generalizing previous results on biharmonic hypersurfaces.

Contribution

It generalizes known results on biharmonic hypersurfaces to biconservative hypersurfaces, including new classifications for $ ext{delta}(2)$- and $ ext{delta}(3)$-ideal cases.

Findings

$ ext{delta}(2)$- and $ ext{delta}(3)$-ideal biconservative hypersurfaces are minimal in Euclidean space.

$ ext{delta}(4)$-ideal biconservative hypersurfaces with constant scalar curvature have constant mean curvature.

Abstract

A biconservative submanifold of a Riemannian manifold is a sub- manifold with divergence free stress-energy tensor with respect to bienergy. These are generalizations of biharamonic submanifolds. In 2013, B. Y. Chen and M.I. Munteanu proved that $\delta(2)$-ideal and $\delta(3)$-ideal biharmonic hypersurfaces in Euclidean space are minimal. In this paper, we generalize this result for $\delta(2)$-ideal and $\delta(3)$-ideal bisonservative hypersurfaces in Euclidean space. Also, we study $\delta(4)$-ideal biconservative hypersurfaces in Euclidean space $\mathbb{E}^{6}$ having constant scalar curvature. We prove that such a hypersurface must be of constant mean curvature.