Prolonged analytic connected group actions are generically free
In the article under review, the authors prove that an effective, analytic action of a connected Lie group \(G\) on an analytic manifold \(M\) becomes free on a comeager subset of an open subset of \(M\) when prolonged to a frame bundle of sufficiently high order. Next, they show that the action of \(G\) becomes free on a comeager subset of an open subset of a submanifold jet bundle over \(M\) of sufficiently high order, thereby establishing a general result that underlies Lie's theory of symmetry groups of differential equations and the equivariant method of moving frames. In Section 1 the topics are introduced and their relevance in Lie's theory of symmetry groups of differential equations and differential invariants, in Cartan's method of moving frames and its equivariant generalization is pointed out. An important question in the moving frame construction is whether the prolonged transformation group action is free on submanifold jet bundles of sufficiently high order. In Section 2 the authors present the basic terminology and constructions for Lie group actions and their prolongations to submanifold jet bundles. Section 3 contains lemmas concerning abelian Lie groups. Section 4 is devoted to the study of meager subsets of topological spaces and their behavior under maps. In Section 5 the first main result, Theorem 20, is stated and proved. Section 6 contains a key technical lemma which states that if a local diffeomorphism fixes the \(n\) jets of all submanifolds passing through a point \(z \in M\), then it has the same \(k\) jet as the identity map for some smaller \(k\) upon which \(n\) depends. In the last Section, the authors prove, in Theorem 26, a similar freeness result for prolonged connected, analytic, effective group actions on submanifold jet bundles, except here the freeness is established on a comeager subset of an open subset of a sufficiently high order jet bundle.
scientific article; zbMATH DE number 6991402
scientific article