TITLE:
Deormation Theory of Geometric Objects
AUTHORS:
Jean-Francois Pommaret
KEYWORDS:
Lie Group, Lie Pseudogroup, Lie Algebroid, Differential Sequence, Geometric Object, Integrability Conditions, Compatibility Conditions, Differential Galois Theory, Structure Constants, Deformation Cohomology, Centralizer, Normalizer
JOURNAL NAME:
Advances in Pure Mathematics,
Vol.16 No.8,
August
31,
2026
ABSTRACT: About at the same time, E. Vessiot (1903) and E. Cartan (1904), the only two French heirs of S. Lie in competition, discovered structure equations that they used to study infinite groups of transformations of a manifold, now called Lie pseudogroups. Using exterior calculus on jet-bundles, Cartan could not quotient down his equations on the manifold, a result only obtained in 1970 by D.C. Spencer with the nonlinear Spencer sequences. On the contrary, Vessiot was able to quotient down his results on the manifold. This paper proves and illustrates through many examples the importance of the Vessiot structure equations totally ignored for one century, particularly by D.C. Spencer or E. Kolchin and their successors. It happens that, apart from the well-known Maurer-Cartan equations for Lie groups, the only Vessiot structure constant known today is that of the constant Riemannian curvature introduced by L. P. Eisenhart in 1918, confusing integrability conditions (IC) with compatibility conditions (CC) in the study of the Killing equations for a given nondegenerate metric but no reference can be found in the whole mathematical literature today. We shall prove that, in this case, two Vessiot structure constants indeed exist, one vanishing and the other being the only known one or both being equal. We also associate with the Vessiot structure constants a new algebraic cohomology generalizing the Chevalley-Eilenberg cohomology of finite dimensional Lie algebras, even if the Lie pseudogroup is infinite dimensional. This new tool is particularly useful for studying the famous equivalence problem for geometric objects and structures on manifolds while computing the finite co-dimension of a transitive Lie pseudogroup in its normalizer that can be used in the Differential Galois Theory (DGT). Most examples can be treated by using computer algebra.