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.