A remark on ${\mathcal{C}}^{\infty }$ definable equivalence

Abstract

We establish that if a submanifold $M$ of ${\mathbb{R}}^n$ is definable in some o-minimal structure then any definable submanifold $N\subset {\mathbb{R}}^n$ which is ${\mathcal{C}}^{\infty }$ diffeomorphic to $M$, with a diffeomorphism $h:N\to M$ that is sufficiently close to the identity, must be ${\mathcal{C}}^{\infty }$ definably diffeomorphic to $M$. The definable diffeomorphism between $N$ and $M$ is then provided by a tubular neighborhood of $M$.

Final version