Biconservative hypersurfaces in space forms with harmonic curvature

Abstract

A biconservative hypersurface $M$ in a space form ${\overline{M}}^{n+1}(c)$ is a hypersurface whose normal part of the vector $\Delta \vec{H}$ vanishes. We study such hypersurfaces with harmonic curvature or parallel Ricci curvature, and prove that have constant mean curvature. Also, by combining our results with previous results of H.B. Lawson Jr. and M. Umehara we obtain that if $M$ is a biconservative hypersurface in a space form ${\overline{M}}^{n+1}(c)$ with parallel Ricci tensor or harmonic curvature, then the shape operator $\mathbf{A}$ of $M$ is covariant constant, i.e. $\nabla \mathbf{A}=0$.

arXiv comment15 pages