Continuous isomorphisms between groups definable in o-minimal expansions of the real field

In this paper we study the relation between the category of real Lie groups and that of groups definable in o-minimal expansions of the real field, which we will refer to as ``definable groups''. With this terminology, it is known (\cite{Pi88}) that any definable group is a Lie group, and in \cite{COP} a complete characterization of when a Lie group is \emph{Lie isomorphic} to a definable group'' was given. We continue the analysis by explaining when a Lie isomorphism between definable groups is definable. Among other things, we generalize Wilkie's result on the o-minimality of the exponential function (\cite{Wilkie}) by completely characterizing when, given an o-minimal expansion $\mathcal R$ of the real field and a Lie isomorphisms $\phi$ between two $\mathcal R$-definable groups $G_1, G_2$, $\phi$ can be added to the language of $\mathcal R$ preserving o-minimality. We also prove that any definable group $G$ can be endowed with an analytic manifold structure definable in $\mathcal R_{\text{Pfaff}}$ that makes it an analytic group.