Ricci curvature bounds and rigidity for non-smooth Riemannian and semi-Riemannian metrics

TL;DR

This paper extends rigidity results like the Cheeger-Gromoll splitting theorem to low-regularity Riemannian and semi-Riemannian metrics, establishing new identities and regularity results in non-smooth geometric analysis.

Contribution

It proves a version of the splitting theorem for low-regularity metrics and introduces a Bochner-Weitzenböck identity applicable to non-smooth settings.

Findings

Established a splitting theorem for $C^1$ metrics.

Derived a Bochner-Weitzenböck identity for non-smooth metrics.

Discussed Sobolev space notions and Ricci curvature bounds in low regularity.

Abstract

We study rigidity problems for Riemannian and semi-Riemannian manifolds with metrics of low regularity. Specifically, we prove a version of the Cheeger-Gromoll splitting theorem \cite{CheegerGromoll72splitting} for Riemannian metrics and the flatness criterion for semi-Riemannian metrics of regularity $C^1$. With our proof of the splitting theorem, we are able to obtain an isometry of higher regularity than the Lipschitz regularity guaranteed by the $\mathsf{RCD}$-splitting theorem \cite{gigli2013splitting, gigli2014splitoverview}. Along the way, we establish a Bochner-Weitzenb\"ock identity which permits both the non-smoothness of the metric and of the vector fields, complementing a recent similar result in \cite{mondino2024equivalence}. The last section of the article is dedicated to the discussion of various notions of Sobolev spaces in low regularity, as well as an alternative proof of the equivalence (see \cite{mondino2024equivalence}) between distributional Ricci curvature bounds and $\mathsf{RCD}$-type bounds, using in part the stability of the variable $\mathsf{CD}$-condition under suitable limits \cite{ketterer2017variableCD}.