Sufficient conditions for a curvature tensor to be Riemannian and for determining its metric

Abstract

A simple necessary and sufficient condition that a curvature tensor be Riemannian (i.e., that its symmetric connection be metric) is found, in very general conditions, for a four-dimensional space-time manifold. This enables existing uniqueness results for the metric tensor and an associated algebraic procedure for obtaining the metric tensor components from the curvature tensor components (when the curvature tensor is known to be a Riemannian tensor), to be extended to the situation where it is not known a priori whether the curvature tensor has come from a Lorentz metric.