  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}.
