Metrisability of three-dimensional path geometries

TL;DR

This paper investigates conditions under which a three-dimensional projective structure can be derived from a metric, identifying explicit invariants that obstruct such metrisability.

Contribution

The authors derive explicit projectively invariant obstructions to the existence of a Levi-Civita connection within a projective class on 3D manifolds.

Findings

Obstructions are given by tensors constructed from the projective Weyl curvature.

Vanishing of these tensors is necessary but not sufficient for metrisability.

Examples demonstrate the limitations of the obstructions.

Abstract

Given a projective structure on a three-dimensional manifold, we find explicit obstructions to the local existence of a Levi-Civita connection in the projective class. These obstructions are given by projectively invariant tensors algebraically constructed from the projective Weyl curvature. We show, by examples, that their vanishing is necessary but not sufficient for local metrisability.