Determination of the metric from the connection

As is well known, a metric on a manifold determines a unique symmetric connection for which the metric is parallel: the Levi-Civita connection. In this paper we investigate the inverse problem: to what extent is the metric of a Riemannian manifold determined by its Levi-Civita connection? It is shown that for a generic Levi-Civita connection of a metric $h$ there exists a set of positive semi-definite tensor fields $h_{a}$ such that the parallel metrics are the positive-definite linear combinations of the $h_{a}$. Moreover, the set of all parallel metrics may be constructed by a soley algebraic procedure.