Some Remarks on Curvature Tensor Integrability in Spacetimes

If a vector field is covariantly constant then the Ricci identity shows that it annihilates the curvature tensor and all its covariant derivatives. This paper attempts a 'best possible' converse to this result in the four-dimensional Lorentz case (the spacetime of general relativity theory) by supposing a vector field annihilates the curvature tensor and a certain number of its derivatives and showing how this leads to a covariantly constant vector field. A direct proof of these results, together with a brief description of how the proof can be obtained using holonomy theory, is presented. The two- and three-dimensional cases are also remarked upon.