Integrability of Lie equations and pseudogroups (Q1840785)

The most fundamental concepts of Lie-Cartan pseudogroup theory are introduced and thoroughly discussed: equivalence between Lie and Cartan approaches (Lie: the transformations satisfying a special system of differential equations, Cartan: the transformations preserving a family of differential forms), the role of infinitesimal transformations of the pseudogroup (leading to so called regular pseudogroups), the Goldschmidt integrability of nonlinear Lie equations (which provide the transformations) and of linear Lie equations (they determine the infinitesimal transformations), and a modification of the first and of the second fundamental Lie's theorem. No examples. The article is of introductory nature. A rather precise approach based on the Weil's theory of near points on a smooth manifold is applied.