Lie group structures on groups of diffeomorphisms and applications to CR manifolds

Lie group structures on automorphism groups of real-analytic CR manifolds

“…For the case of Levi-degenerate CR manifolds, the same conclusion was recently obtained by Baouendi, Rothschild, Winkelmann and the third author [BRWZ04] for the class of finitely nondegenerate minimal CR manifolds, which corresponds here to our Theorem 1.2 with K = ∅. (We should point out that the results in those mentioned papers also apply for merely smooth CR manifolds as well, based on the previous work [KZ05], but in this paper we shall focus on the real-analytic category.…”

“…Hence, Theorem 1.2 applied with K := S 1 × {0}, yields that Aut CR (M) is a Lie group. On the other hand, M is not finitely nondegenerate at any point of K and hence the results from [BRWZ04] do not apply to M.…”

“…We now fix an arbitrary set S of germs of local real-analytic diffeomorphisms h : (M, p) → M with possibly varying reference point p ∈ M and, as in [BRWZ04], consider the following condition.…”

“…Our proof of Theorem 1.2 makes use of the recent developments providing a relationship between various notions and results concerning jet parametrization of local CR diffeomorphisms [BER97, Z97, BER99b, E01, KZ05, LM05] and Lie group structures on (local and) global groups of automorphisms of CR manifolds [BRWZ04]. In the next section, we give in Theorem 2.2 new sufficient conditions on a connected real-analytic CR manifold M, in terms of local jet parametrization properties of CR automorphisms, that ensure that Aut CR (M) is a Lie group.…”

“…The compact-open C r topology on Aut CR (M) is now induced by the appropriate topology relative to the coordinate charts for the maps and their inverses (see e.g. [BRWZ04] for a more detailed discussion). For brevity, we adopt the order k < ∞ < ω for any integer k.…”