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

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 {CheegerGromoll72splitting} 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 {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 {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 {mondino2024equivalence}) between distributional Ricci curvature bounds and RCD-type bounds, using in part the stability of the variable CD-condition under suitable limits {ketterer2017variableCD}.