Totally umbilical submanifolds in pseudo-Riemannian space forms

TL;DR

This paper classifies full totally umbilical submanifolds in non-flat pseudo-Riemannian space forms and explores their moduli spaces, revealing non-Hausdorff properties in certain isometric immersion classes.

Contribution

It provides a classification of congruence classes of full totally umbilical submanifolds and analyzes the structure of their moduli spaces in pseudo-Riemannian space forms.

Findings

Classification of full totally umbilical submanifolds in non-flat pseudo-Riemannian space forms.

Identification of non-Hausdorff moduli spaces for certain isometric immersions.

Insights into the geometric structure of submanifolds in pseudo-Riemannian geometry.

Abstract

A totally umbilical submanifold in pseudo-Riemannian manifolds is a fundamental notion, which is characterized by the condition that the second fundamental form is proportional to the metric. It is also a generalization of the 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 their moduli spaces. As a consequence, we show that some moduli spaces of isometric immersions between space forms which one of the same constant curvature are non-Hausdorff.