Let $M$ be a connected and simply connected Riemannian manifold, non-compact, and suppose that $\{V_p\}_{p \in M}$ is a family of vector fields on $M$ indexed by $M$. Suppose moreover that the map $$ p\to V_p(x), $$ is smooth and define the differential equation $$ f_t' = V_p(f_t) ,\, f_0=p. \qquad (*) $$ So in $(*)$ there is an explicit dependence on $p$ both in the initial condition (IC) and in the "dynamics" of the ODE itself. Therefore, let's relabel this explicit dependence on $p$ of a solution to $(*)$ by $f_{t,p}$.
Let $X$ be the collection of all compactly-supported smooth vector-fields on $M$ such that
- A solution $f_{t,p}$ exists up to time $1$ for each $p$,
- The map taking $(V_p)_{p \in M}$ to the time $1$ solution of $(*)$ is a diffeomorphism,
- $p\to V_p(x)$ is non-constant for at-least one value of $x\in M$.
Denote this latter map by $\widetilde{\operatorname{Exp}}:X\to \operatorname{Diff}_{c,0}(M)$. It is known, see this post for references, that if there is no dependence on $p$ then $\operatorname{Exp}$ generates $\operatorname{Diff}_{c,0}(M)$. Are such results known in this case, i.e.: where vector fields depend non-trivially on the ICs?
