Riemannian Geometry of $C^{1,1}$ Manifolds

TL;DR

This paper develops Riemannian geometry for $C^{1,1}$ manifolds, highlighting differences from classical $C^{2}$ cases, including non-symmetric connections and additional curvature terms affecting fundamental geometric and physical equations.

Contribution

It introduces a framework for Riemannian geometry at the $C^{1,1}$ regularity level, accounting for non-symmetric connections and their impact on curvature and geodesic equations.

Findings

Connection is not symmetric in $C^{1,1}$ manifolds.

Additional terms appear in curvature tensors and geodesic equations.

Failure to include these terms results in nonzero torsion.

Abstract

Riemannian Geometry for $C^{1,1}$ manifolds contains important differences from that for $C^{2}$ manifolds. This paper develops Riemannian geometry at the $C^{1,1}$ level of regularity. It is shown that the connection is not symmetric and this leads to additional terms in curvature tensors, geodesic equations and the Bianchi identities. Failure to account for these terms leads to nonzero torsion, affecting everything from geodesics to the Einstein curvature tensor.