2002/12/12 by P Dolan, Patrick Dolan, Dolan, P +3 · 1 citation
Mathematics · Physics and Astronomy · #Advanced Differential Equations and Dynamical Systems #FOS: Physical sciences #General Relativity and Quantum Cosmology (gr-qc) #Nonlinear Waves and Solitons #Quantum chaos and dynamical systems #gr-qc
paper · pdf · doi:10.48550/arxiv.gr-qc/0212054
26 pages, latex, no figures
arxiv created 2002/12/12 · openalex publication_date 2002/12/12 · arxiv updated 2009/11/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
The key problem of the theory of exterior differential systems (EDS) is to decide whether or not a system is in involution. The special case of EDSs generated by one-forms (Pfaffian systems) can be adequately illustrated by a 2-dimensional example. In 4 dimensions two such problems arise in a natural way, namely, the Riemann-Lanczos and the Weyl-Lanczos problems. It is known from the work of Bampi and Caviglia that the Weyl-Lanczos problem is always in involution in both 4 and 5 dimensions but that the Riemann-Lanczos problem fails to be in involution even for 4 dimensions. However, singular solutions of it can be found. We give examples of singular solutions for the Goedel, Kasner and Debever-Hubaut spacetimes. It is even possible that the singular solution can inherit the spacetime symmetries as in the Debever-Hubaut case. We comment on the Riemann-Lanczos problem in 5 dimensions which is neither in involution nor does it admit a 5-dimensional involution of Vessiot vector fields in the generic case.