Biharmonic submanifolds of pseudo-Riemannian manifolds

TL;DR

This paper develops biharmonic equations for pseudo-Riemannian submanifolds, classifies certain biharmonic hypersurfaces, and explores their properties and constructions, extending known Riemannian results to the pseudo-Riemannian setting.

Contribution

It introduces biharmonic equations for pseudo-Riemannian submanifolds and classifies biharmonic hypersurfaces with limited principal curvatures, providing new construction methods.

Findings

Pseudo-umbilical biharmonic submanifolds have constant mean curvature

Complete classification of biharmonic hypersurfaces with up to two principal curvatures

Four methods to construct proper biharmonic submanifolds from minimal ones

Abstract

In this paper, we derived biharmonic equations for pseudo-Riemannian submanifolds of pseudo-Riemannian manifolds which includes the biharmonic equations for submanifolds of Riemannian manifolds as a special case. 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.