Prolongation methods and Cartan characters for the three-dimensional Riemann–Lanczos problem
Annelies Gerber · Journal of Mathematical Physics · 2003
The Riemann–Lanczos problem in two dimensions is in involution and its general solution is known. The three-dimensional Riemann–Lanczos problem is not in involution and needs to be prolongated. We write the three-dimensional Riemann–Lanczos problem as an exterior differential system and prolong it to second order. Using algebraic computing we find that we have to add an integrability condition to make it a system in involution. We also suggest a prolongation of the three-dimensional Riemann–Lanczos problem in the same way as Bampi and Caviglia did for four dimensions. After supplementing this three-dimensional system with an integrability condition, it becomes involutive with Cartan characters (17,8,2) or (20,10,3) if no cyclic conditions are imposed. We present the relevant sections of REDUCE computer codes by means of which these results were obtained.