It is well known that a group definable in an o-minimal structure can be definably endowed with a differential manifold structure that makes the group operations smooth (a Lie group).
If \(G_1\) and \(G_2\) are two definable groups in an o-minimal expansion \(\mathcal R\) of the real field with a smooth isomorphism \(\phi\) between them, when is \(\phi\) definable in \(\mathcal R\)?
We will sketch the proof that, as with the exponential function, if \(\phi\) is notdefinable then adding \(\phi\) preserves o-minimality of \(\mathcal R\). We will then talk about what is known about outright definability (and not) of \(\phi\).