Exterior differential systems, Janet–Riquier theory and the Riemann–Lanczos problems in two, three, and four dimensions

Abstract

We discuss the Riemann–Lanczos problems in two, three, and four dimensions using the theory of exterior differential systems and Janet–Riquier theory. We show that the Riemann–Lanczos problem in two dimensions is always a system in involution. For each of the two possible signatures we give the general solution in both instances and show that the occurrence of characteristic coordinates need not affect the result. In three dimensions, the Riemann–Lanczos problem is not in involution as an identity occurs. This does not prevent the existence of singular solutions and we give an example for the reduced Gödel space–time. A prolongation of this problem, whereby an integrability condition is added, leads to a prolonged system in involution. The Riemann–Lanczos problem in four dimensions is not in involution and needs to be prolongated as Bampi and Caviglia suggested. But singular solutions of it can be found and we give examples for the Gödel, Kasner, and Debever–Hubaut space–times.