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

2024/06/10 by Kunzinger, Michael, Ohanyan, Argam, Vardabasso, Alessio · 1 citation
#46T30 #49Q22 #53C21 #53C24 #Differential Geometry (math.DG) #FOS: Mathematics #Metric Geometry (math.MG)

paper · doi:10.48550/arxiv.2406.06762

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 \citeCheegerGromoll72splitting for Riemannian metrics and the flatness criterion for semi-Riemannian metrics of regularity C¹. With our proof of the splitting theorem, we are able to obtain an isometry of higher regularity than the Lipschitz regularity guaranteed by the RCD-splitting theorem \citegigli2013splitting, gigli2014splitoverview. Along the way, we establish a Bochner-Weitzenböck identity which permits both the non-smoothness of the metric and of the vector fields, complementing a recent similar result in \citemondino2024equivalence. 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 \citemondino2024equivalence) between distributional Ricci curvature bounds and RCD-type bounds, using in part the stability of the variable CD-condition under suitable limits \citeketterer2017variableCD.

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• Equivalence of the null energy condition to variable lower bounds on the timelike Ricci curvature for C²-Lorentzian metrics

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