On the curvature description of gravitational fields

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Abstract
The local geometry of a Riemannian manifold is described completely by the curvature tensor and a finite number of its covariant derivatives. The authors derive necessary and sufficient conditions for a line element to exist that gives rise to these quantities. The resulting system of equations can be written as a set of integrable differential equations along with a set of algebraic equations. This gives a technique for searching for solutions to Einstein's equations with special properties. It also makes it possible to perform perturbative calculations entirely in terms of invariant quantities. They illustrate the methods on homogeneous rotationally symmetric spacetimes.