Virtual immersions, and a characterization of symmetric spaces

TL;DR

This paper introduces the concept of virtual immersions as a generalization of isometric immersions, characterizes symmetric spaces via skew symmetric second fundamental forms, and establishes their uniqueness.

Contribution

It defines virtual immersions, links skew symmetric second fundamental forms to symmetric spaces, and proves their essential uniqueness.

Findings

Manifolds admit virtual immersions with skew symmetric second fundamental form iff they are symmetric spaces.

Virtual immersions have a second fundamental form that is generally not symmetric.

Such virtual immersions are essentially unique for symmetric spaces.

Abstract

We define virtual immersions, as a generalization of isometric immersions in a pseudo-Riemannian vector space. We show that virtual immersions possess a second fundamental form, which is in general not symmetric. We prove that a manifold admits a virtual immersion with skew symmetric second fundamental form, if and only if it is a symmetric space, and in this case the virtual immersion is essentially unique.