Metrizability of connections on two-manifolds

summary:We contribute to the reverse of the Fundamental Theorem of Riemannian geometry: if a symmetric linear connection on a manifold is given, find non-degenerate metrics compatible with the connection (locally or globally) if there are any. The problem is not easy in general. For nowhere flat $2$-manifolds, we formulate necessary and sufficient metrizability conditions. In the favourable case, we describe all compatible metrics in terms of the Ricci tensor. We propose an application in the calculus of variations.