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Lie group structures on groups of diffeomorphisms and applications to CR manifolds. (English) Zbl 1062.22046
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.

##### MSC:
 22F50 Groups as automorphisms of other structures 57S25 Groups acting on specific manifolds 58D05 Groups of diffeomorphisms and homeomorphisms as manifolds 32V40 Real submanifolds in complex manifolds 22E15 General properties and structure of real Lie groups
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