ABSTRACT
Given a locally integrable structure V over a smooth manifold Ω and given p ∈ Ω we define the Borel map of V at p as the map which assigns to the germ of a smooth solution of V at p its formal Taylor power series at p. In this work we continue the study initiated in Barostichi et al. (Math. Nachr. 286(14-15):1439-1451, 2013), Della Sala and Lamel (Int J Math 24(11):1350091, 2013) and present new results regarding the Borel map. We prove a general necessary condition for the surjectivity of the Borel map to hold and also, after developing some new devices, we study some classes of CR structures for which its surjectivity is valid. In the final sections we show how the Borel map can be applied to the study of the algebra of germs of solutions of V at p.
ABSTRACT
Let G be a unimodular Lie group, X a compact manifold with boundary, and M the total space of a principal bundle [Formula: see text] so that M is also a strongly pseudoconvex complex manifold. In this work, we show that if there exists a point [Formula: see text] such that [Formula: see text] is contained in the complex tangent space [Formula: see text] of bM at p, then the Bergman space of M is large. Natural examples include the gauged G-complexifications of Heinzner, Huckleberry, and Kutzschebauch.
