Geodesic
Metrizability of connections on two-manifolds
Acta Universitatis Palackianae Olomucensis. Facultas Rerum Naturalium. Mathematica, Tome 48 (2009) no. 1, pp. 157-170.

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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.
Classification : 53B05, 53B20, 53C05
Keywords: Manifold; linear connection; metric connection; pseudo-Riemannian geometry 
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Vanžurová, Alena; Žáčková, Petra. Metrizability of connections on two-manifolds. Acta Universitatis Palackianae Olomucensis. Facultas Rerum Naturalium. Mathematica, Tome 48 (2009) no. 1, pp. 157-170. http://geodesic.mathdoc.fr/item/AUPO_2009__48_1_a13/