1
$\begingroup$

I recently came across a way to think of an ordinary differential equation on a smooth manifold $M$ is as a Lie group homomorphism $\phi : (\mathbb{R}, +) \rightarrow \operatorname{Diff}(M)$ where $\operatorname{Diff}(M)$ is the group of smooth diffeomorphisms of $M$.

If we substitute the additive group of reals with another Lie group $G$ in this point of view, we could regard the resulting map $\phi : G \rightarrow \operatorname{Diff}(M)$ as a "$G$-differential equation" or "$G$-flow" on $M$.

Do these kinds of flows naturally arise and are they well studied? Are there some nice examples of $G$-flows for classical groups?

Improve this question
$\endgroup$
5
  • $\begingroup$ It's not quite precisely what you want, but I think you're looking at things along the lines of the moment map. $\endgroup$
    – user44191
    Commented Nov 27, 2021 at 1:59
  • 7
    $\begingroup$ I would say that what you are calling a ``$G$-flow'' is usually known as a smooth action of $G$ on $M$. Yes, there are plenty of these. Although $Diff(M)$ is not really a Lie group, it does have a Lie algebra associated with it, namely the space $\Gamma (TM)$ of smooth tangent fields. A smooth $G$-action gives a Lie algebra map to $\Gamma (TM)$, and if $G$ is connected then the Lie algebra map determines the action. $\endgroup$
    – Tom Goodwillie
    Commented Nov 27, 2021 at 4:37
  • 1
    $\begingroup$ The definition of a group action is standard; for Lie group actions, one only requires smoothness. Look in any undergraduate differential geometry textbook that covers Lie groups. $\endgroup$
    – Ben McKay
    Commented Nov 27, 2021 at 7:08
  • 2
    $\begingroup$ @Tom Goodwillie: What do you mean by Diff$(M)$ is not really a Lie group? ITS a classical result that the diffeomorphism group is an Infinite dimensional Lie group. $\endgroup$
    – Alexander Schmeding
    Commented Nov 27, 2021 at 8:09
  • 2
    $\begingroup$ @Alexander Schmeding: I mean, while it can be useful to generalize the idea of Lie group to some infinite-dimensional cases, it will not be true that all the theorems of Lie group theory extend. $\endgroup$
    – Tom Goodwillie
    Commented Nov 28, 2021 at 12:37

0

Reset to default