Totally umbilical submanifolds in pseudo-Riemannian space forms

Abstract

A totally umbilical submanifold in pseudo-Riemannian manifolds is a fundamental notion, which is characterized by the fact that the second fundamental form to be proportional to the metric. And, it is also a generalization of a notion of a totally geodesic submanifold. In this paper, we classify congruent classes of full totally umbilical submanifolds in non-flat pseudo-Riemannian space forms, and consider its moduli spaces. As a consequence, we obtain that some moduli spaces of isometric immersions between space forms whose curvatures have the same constant are non-Hausdorff.