Determination of the Metric from the Connection

A metric on a manifold defnes a unique symmetric connection for which the metric is parallel: the Levi-Civita connection. In this paper we investigate the inverse problem: the determination of all parallel metrics of a given Riemannian manifold. It is known that for a generic Levi-Civita connection of a metric h there exists a set of positive semi-defnite metrics ha such that the parallel metrics are the positive-defnite linear combinations of the ha. The proof of this result however, is existential, depending upon solutions to differential equations. In this paper we provide a constructive algebraic approach to the problem.