Janet-Riquier Theory and the Riemann-Lanczos Problems in 2 and 3 Dimensions

The Riemann-Lanczos problem for 4-dimensional manifolds was discussed by Bampi and Caviglia. Using exterior differential systems they showed that it was not an involutory differential system until a suitable prolongation was made. Here, we introduce the alternative Janet-Riquier theory and use it to consider the Riemann-Lanczos problem in 2 and 3 dimensions. We find that in 2 dimensions, the Riemann-Lanczos problem is a differential system in involution. It depends on one arbitrary function of 2 independent variables when no differential gauge condition is imposed but on 2 arbitrary functions of one independent variable when the differential gauge condition is imposed. For each of the two possible signatures we give the general solution in both instances to show that the occurrence of characteristic coordinates need not affect the result. In 3 dimensions, the Riemann-Lanczos problem is not in involution as a identity occurs. This does not prevent the existence of singular solutions. A prolongation of this problem, where an integrability condition is added, leads to an involutory prolonged system and thereby generates non-singular solutions of the prolonged Riemann-Lanczos problem. We give a singular solution for the unprolonged Riemann-Lanczos problem for the 3-dimensional reduced Goedel spacetime.