Biharmonic Submanifolds and Functions of Generalized Space Form and Pseudo-Riemannian Manifolds and on Special Unitary Group

Abstract

A Survey on biharmonic maps between Riemannian manifolds and a classification on marginally trapped Lorentzian flat surfaces are studied. The biharmonic submanifolds in nonflat Lorentz 3-space forms, with parallel mean curvature, of generalized space forms and of pseudo-Riemannian manifolds are obtained. We characterize the biharmonic hypersurfaces in Riemannian manifolds, of 4-dimensional semi-Euclidean space and proper biharmonic hypersurfaces in pseudo-Riemannian space form. The harmonic morphisms between semi-Riemannian manifolds, from the Grassmannians and their non-compact duals, and from compact semi-simple Lie groups are investigated. The biharmonic functions on the classical compact simple Lie groups and the special unitary group are discussed.