First of all, we have to give the group $\text{Diff}(M)$ of all diffeomorphisms on $M$ a smooth-manifold structure. (To see this, it may be helpful to consider a easier problem: how to give the group $\text{Homeo}(M)$ of all homeomorphisms on $M$ a topology structure. I guess it is possibly related to compact-open topology on the sets of continuous maps between two topological spaces.)

In addition, the local flow $\phi_t$ of a smooth vector field $X\in \mathcal X(M)$ may be a good example of a smooth family of diffeomorphisms. With this in mind, I was wondering if the following statement is a equivalent condition of the smoothness of a family $\phi_t\in \text{Diff}(M)$: for every $p \in M$ the path $t \mapsto \phi_t(p)$ is smooth in $M$.

I am reading Mcduff & Salamon's book on symplectic topology. The following text and proposition is the place where I encounter my problem (Page 83). enter image description here


I would expect that for the purpose you are aiming at it is sufficient that $\psi$ is smooth as a map $[0,1]\times M \rightarrow M$ (where the first factor has a smooth structure in a natural way) with the additional constraint that, for each $t$, $\psi_t $ is a diffeomorphism (or symplectomorphism). This is, however, not the structure of a differentiable manifold on $\text{Diff}(M)$, and for most cases this is not needed.

A differential structure on the set of Diffeomorphisms is useful, however, if you intent to prove smooth dependency of the flow on initial conditions (or in general, smooth dependency of solutions of ODE from their initial conditions). This is done, e.g, in Serge Lang's Real Analysis, using (an infinite dimensional version of) the implicit function theorem.


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