Biharmonic submanifolds of pseudo-Riemannian manifolds

Abstract

In this paper, we derived biharmonic equations for pseudo-Riemannian submanifolds of pseudo-Riemannian manifolds which include the biharmonic equations for submanifolds of Riemannian manifolds as special cases. As applications, we proved that a pseudo-umbilical biharmonic pseudo-Riemannian submanifold of a pseudo-Riemannian manifold has constant mean curvature, we completed the classifications of biharmonic pseudo-Riemannian hypersurfaces with at most two distinct principal curvatures, which were used to give four construction methods to produce proper biharmonic pseudo-Riemannian submanifolds from minimal submanifolds. We also made some comparison study between biharmonic hypersurfaces of Riemannian space forms and the space-like biharmonic hypersurfaces of pseudo-Riemannian space forms.