On standard Riemannian space forms and their c-biconservative hypersurfaces

Abstract

According to a variational problem, the tensor of stress-energy, as specified by Hilbert (1924), is a bicovariant symmetric tensor with null divergence. This property is named the conservativeness of stress-energy tensor. In this literature, the stress-energy tensor associated to the bi-energy function with null divergence is said to be biconservative. In differential geometric point of view, a hypersurface $\xi : M^n\rightarrow\M^{n+1}(c)$ of a Riemannian space form is called biconservative if $\Delta^2\xi$ has null tangential component, where $\Delta$ is the Laplace operator on $M^n$. It is proved that such a hypersurface has constant mean curvature. We consider the hypersurfaces satisfying a progressive version of biconservativity condition. The $\Box$-biconservativity condition is obtained by substituting the Cheng-Yau operator $\Box$ instead of $\Delta$. We prove that $\Box$-biconservative hypersurfaces of Riemannian $(n+1)$-space forms (with some additional conditions) have constant scalar curvature.