A remark on $\mathscr C^\infty $ definable equivalence
Volume 131 / 2023
Annales Polonici Mathematici 131 (2023), 79-84
MSC: Primary 32B20; Secondary 58C25, 03C64.
DOI: 10.4064/ap230321-10-8
Published online: 6 September 2023
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 $\mathscr C^\infty $ diffeomorphic to $M$, with a diffeomorphism $h:N\to M$ that is sufficiently close to the identity, must be $\mathscr C^\infty $ definably diffeomorphic to $M$. The definable diffeomorphism between $N$ and $M$ is then provided by a tubular neighborhood of $M$.
