Sufficiency of the Ricci equations for characterizing the Riemann tensor (Q803545)

Suppose one has a tensor \(K^ a_{bcd}\) on a space-time manifold M with Lorentz metric g which has the algebraic symmetries of a Riemann tensor \((K^ a_{bcd}=-K^ a_{bdc}\text{ and } K^ a_{[bcd]}=0)\) and satisfies the relation \(g_{ac}K^ e_{bcd}=-g_{be}K^ e_{acd}\) and the Ricci identity (called the Ricci equations in this paper) with respect to the Levi-Civita connection arising from g (but with K replacing the Riemann tensor in the usual expression). Is K then the Riemann tensor arising from the Levi-Civita connection of g? This paper addresses itself to this problem, following an earlier paper [ibid. 6, No.1, 159-163 (1989; Zbl 0681.53007)] by \textit{A. D. Rendall} who considered a similar problem. The author shows that if K satisfies one of a set of four conditions that it is necessarily the Riemann tensor associated with g. The proof consists of constructing the tensor \(P^ a_{bcd}\equiv K^ a_{bcd}- R^ a_{bcd}\) (given in Rendall's paper above) and using an algebraic decomposition of K (similar to a standard one used for Riemann tensors) to show that any of the above mentioned four conditions leads to \(P=0\). The reader who wishes to check details should note that on page 193 of this paper, line 16, the equation/reference (3) should read (2)! The author proceeds to show that the above sufficient conditions for K to be the Riemann tensor of g are generic (using a Whitney type topological argument based on Rendall's paper). An appendix is given which contains the calculations for the algebraic part of the argument.