Metrisability of two-dimensional projective structures

Abstract

We carry out the programme of R. Liouville, Sur les invariants de certaines équations différentielles et sur leurs applications, to construct an explicit local obstruction to the existence of a Levi–Civita connection within a given projective structure $\Gamma$ on a surface. The obstruction is of order 5 in the components of a connection in a projective class. It can be expressed as a point invariant for a second order ODE whose integral curves are the geodesics of $\Gamma$ or as a weighted scalar projective invariant of the projective class. If the obstruction vanishes we find the sufficient conditions for the existence of a metric in the real analytic case. In the generic case they are expressed by the vanishing of two invariants of order 6 in the connection. In degenerate cases the sufficient obstruction is of order at most 8.