Suppose $M$ be a smooth manifold with some conditions/structures (1). For instance, metric, holomorphic structure, etc..
Then, let $X$ be a nowhere vanishing vector field that respects the (1) of $M$. (By Lie derivative, or parallele transport, etc.) We now call such vector fields as (2) (If exists).
If we collect the flows of vector fields under (2) iniated from each point in $M$ and makes a space $\mathcal{F}$, then what is the known good properties of $\mathcal{F}$ under the assumption (1), (2)? Are there some good categories to unify them in one foundation?
It's kind of quotient problem, because all of $x$ and $y$ in $M$ is identified by the above vector flows. But, it's not quite same with the situation in quotient manifold theorem, because there may be singular(or periodic) orbits, or even worse, they can be immersed, or densely filling the space. Hence, they are not Hausdorff, neither, in some situation.
At this moment, We can ask also, that there is some good properties on (2) : it can be chosen that their flows are not densely filling the space. i.e., asking the possibilities on perturbation. We can call such classes, (3)
I've seen the similar concenpts before, maybe in symplectic quotient, or something, but I'm not sure.
I'm sorry about asking this too big and general question, since (1) and (2) (and the possibility of (3)?) can be arbitrary, but I'm really curious how many results are there about the spaces of flow.
