$\DeclareMathOperator\Lip{Lip}X$ and $Y$ are compact metric spaces. $\Lip((X,d), (Y,\rho))$ is all Lipschitz maps from $X$ to $Y$.
Is there a topology on $\Lip((X,d), (Y,\rho))$ that makes it a Baire space, closed subspaces Baire and composition continuous?
What if $Y$ is a Riemannian manifold with the associated topological metric $\rho$, same questions and is $\Lip((X,d), (Y,\rho))$ a Banach manifold?
Is Rademacher's theorem (Lipschitz functions are differentiable almost everywhere) true for maps between Riemannian manifolds using the associated topological metrics and a measure on the domain coming from a smooth volume form.
These seem natural and elementary questions that someone must have studied but I can't find any relevant literature. This is for a question in dynamics. I have looked through what I thought were the relevant sections of "Lipschitz Functions" by Cobzas, et. al. For question 1, one can isometrically embed $(Y,\rho)$ in a Banach space and use subtraction there to define the Lipschitz part of the usual metric, but I don't see how to make composition continuous. I would think that questions 2 and 3 would follow by standard methods of global analysis (pass to charts, etc) but I haven't worked it through since I am hoping someone else has. For 3. in charts use the fact that a topological metric coming from a Riemannian one is Lipschitz equivalent to the Euclidean one and any measure coming from a smooth volume form is equivalent to Lebesgue and then use the classical theorem.
