**Papers Published**

- with Bryant, R; Dunajski, M; Eastwood, M,
*Metrisability of two-dimensional projective structures*, Journal of Differential Geometry, vol. 83 no. 3 (2009), pp. 465-500, ISSN 0022-040X

(last updated on 2018/10/14)**Abstract:**

We carry out the programme of R. Liouville \cite{Liouville} 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.