Local decompositions of space-time metric structures into flat parts (Q1069136)

The purpose of this paper is to discuss under what conditions the metric \(ds^ 2\) of a space-time M can be represented locally as a finite sum of flat members \(ds^ 2/U\), where U denotes an open subset of M. We quote the following proposition: If \(ds^ 2/U\) is a sum of k-flat metrics, such that for each of them the canonical coordinates exist on the whole of U, then \((U,ds^ 2/U)\) admits an isometric immersion into \((E^{4k},dt^ 2)\) where \(E^{4k}\) is a pseudo-Euclidean space of signature (k,3k). An application in the case of the Schwarzschild solution is also given.