On smooth locally o-minimal functions

Abstract

We study the $\mathcal{C}^{\infty}$-smooth functions which are locally definable in an o-minimal expansion of the real exponential field with some additional smoothness conditions. Here, the local definability generalizes the subanalytic setting to more transcendental sets and functions. The focus is set on the locally definable diffeomorphisms between manifolds, for which we prove analogies to classical differential geometric results. Moreover, we investigate the relation between classical diffeomorphy and locally definable diffeomorphy.