Lie group structures on groups of diffeomorphisms and applications to CR manifolds. (Q1774063)

The authors obtain some sufficient geometric conditions on a CR manifold \(M\) to guarantee that the group of all its smooth CR automorphisms has the structure of a (finite dimensional) Lie group compatible with its natural topology. First, they establish general theorems on Lie group structures for subgroups of diffeomorphisms of a given smooth real-analytic manifold, then they apply these results together with recent work concerning jet parametrization and complete systems for CR automorphisms. The authors define the notion of a complete system for a set of diffeomorphisms of a manifold and show that it is equivalent to the notion of jet parametrization. Similar results are obtained for subsets of germs of diffeomorphisms fixing a point. Some partial converse results are obtained. Next, the authors present the basic definitions and properties of abstract and embedded CR manifolds that guarantee the existence of a Lie group structure on the group of CR automorphisms.