$C^{1,1}$ Pseudohermitian, Torsion-free Manifolds

TL;DR

This paper investigates $C^{1,1}$ pseudohermitian, torsion-free manifolds, showing they are equivalent to Riemannian manifolds under certain regularity conditions, and reformulating Einstein's pseudohermitian condition for such manifolds.

Contribution

It provides a new analysis of $C^{1,1}$ pseudohermitian, torsion-free manifolds, including a reformulation of Einstein's pseudohermitian condition to account for non-tensorial issues.

Findings

$C^{1,1}$ pseudohermitian, torsion-free manifolds are equivalent to Riemannian manifolds.

A reformulation of Einstein's pseudohermitian condition is provided for non-tensorial contexts.

The analysis extends the understanding of low-regularity geometric structures.

Abstract

Riemannian Manifolds may be $C^{1,1}$ and the geometry of these manifolds is investigated in \cite{Groah1}. Here, a similar analysis is given for pseudohermitian, torsion-free manifolds whereby, instead of assuming that the metric is parallel, it is assumed that the metric is pseudohermitian, a condition adopted by Einstein and elaborated upon in \cite{Hlavaty}. At the level of regularity assumed here, Einstein's formulation of the pseudohermitian condition is not tensorial and so a reformulation of this condition is given here. It is shown that a $C^{1,1}$ manifold is pseudohermitian and torsion-free if and only if it is Riemannian.