The Borel map in locally integrable structures

Abstract

Given a locally integrable structure V over a smooth manifold Ω and given p∈ Ω we define the Borel map ofV atp 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. © 2019, The Author(s).