Every strongly definable $C^{r}G$ vector bundle admits a unique strongly definable $C^{infty}G$ vector bundle structure (Model theoretic aspects of the notion of independence and dimension)

Abstract

Let G be a compact subgroup of GL_{n}(mathbb{R}). We prove that every strongly definable C^{r}G vector bundle over an affine definable C^{infty}G manifold admits a unique strongly definable C^{infty}G vector bundle structure up to definable C^{infty}G vector bundle isomorphism (0leqq r<infty).