On torse-forming vector fields and biharmonic hypersurfaces in Riemannian manifolds

Abstract

In this paper, we give some properties of biharmonic hypersurface in Riemannian manifold has a torse-forming vector field. We prove that every biharmonic hypersurface which is not hyperplane in Euclidean space $\mathbb{R}^{m+1}$ equipped with the Riemannian metric $$\langle,\rangle=\frac{1}{u+v \,y_{m+1}^2}\left(dy_1^2+...+dy_{m}^2\right)+dy_{m+1}^2,$$ is harmonic, where $u,v>0$ are constants