Singular Riemannian metrics, sub-rigidity vs rigidity

TL;DR

This paper investigates the rigidity properties of sub-Riemannian and lightlike metrics, revealing that while generally non-rigid, they can exhibit rigidity in generic cases under certain conditions.

Contribution

It provides a formal analysis of the rigidity versus sub-rigidity of these metrics using Cartan's and Gromov's frameworks, highlighting conditions for rigidity.

Findings

Sub-Riemannian and lightlike metrics are generally non-rigid.

In generic cases, these metrics can induce rigid geometric structures.

The study clarifies the circumstances under which these metrics exhibit rigidity.

Abstract

We analyze sub-Riemannian and lightlike metrics from the point of view of their rigidity as geometric structures. Following Cartan's and Gromov's formal definitions, they are never rigid, yet, in generic cases, they naturally give rise to rigid geometric structures!?