Biconservative hypersurfaces in space forms $\overline{M}^{\lowercase{n+1}}(\lowercase{c})$

TL;DR

This paper investigates biconservative hypersurfaces with four distinct principal curvatures in space forms, proving they must have constant mean and scalar curvatures, thus advancing understanding of their geometric properties.

Contribution

It establishes that biconservative hypersurfaces with four principal curvatures and constant second fundamental form norm necessarily have constant mean and scalar curvatures.

Findings

Hypersurfaces have constant mean curvature.

Hypersurfaces have constant scalar curvature.

All such hypersurfaces are characterized by these curvature conditions.

Abstract

In this paper we study biconservative hypersurfaces $M$ in space forms $\overline M^{n+1}(c)$ with four distinct principal curvatures whose second fundamental form has constant norm. We prove that every such hypersurface has constant mean curvature and constant scalar curvature.