Metrisability of three-dimensional path geometries
  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. 
Maciej Dunajski and Michael Eastwood
math.DG hep-th math-ph math.MP