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 [22] 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 $\mathrm{RCD}$ -splitting theorem [30, 31]. 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 [62]. 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 [62]) between distributional Ricci curvature bounds and $\mathrm{RCD}$ -type bounds, using in part the stability of the variable $\mathrm{CD}$ -condition under suitable limits [47].