On the geometry of biconservative submanifolds in pseudo-Riemannian space forms

Abstract

{In this paper, biconservative submanifolds $M^n_r$ with parallel normalized mean curvature vector field (pnmc) in pseudo-Riemannian space forms are studied under the assumption that $M^n_r$ has diagonalizable shape operator. We first study the scalar curvature rigidity of such non-cmc submanifolds due to the seminar papers by S. S. Chern et al. and prove that the scalar curvature is a fractional power polynomial concerning the mean curvature with nonzero constant coefficients. Second, we discuss the nonexistence problem of non-cmc biconservative submanifolds.}