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...

We discuss Riemannian metrics compatible with a linear connection that has regular curvature. Combining (mostly algebraic) methods and results of [4] and [5] we give an algorithm which allows to decide effectively existence of positive definite metrics compatible with a real analytic connection with regular curvature tensor on an analytic connected and simply connected manifold, and to construct the family of compatible metrics (determined up to a scalar multiple) in the affirmative case. We also...

We contribute to the following: given a manifold endowed with a linear connection, decide whether the connection arises from some metric tensor. Compatibility condition for a metric is given by a system of ordinary differential equations. Our aim is to emphasize the role of holonomy algebra in comparison with certain more classical approaches, and propose a possible application in the Calculus of Variations (for a particular type of second order system of ODE’s, which define geodesics of a linear...