Level set as the orbit of the action of a Lie Group?

Question

I'm wondering the following. Given a smooth function $f:\mathbb R^n\rightarrow \mathbb R^m$ with $m<n$ and level sets $\mathcal O(y)=\{x\in\mathbb R^n| f(x)=y \}$.

What are the conditions on $f$ for which $\mathcal O(y)$ are the orbits of a Lie group action on $\mathbb R^n$? I think that also there are necessary conditions on the kind of action, e.g. that it ought to be proper?

Answer 1

I think the question is probably too broad to have a very satisfying general answer. But (assuming you're looking for a smooth Lie group action) here are a couple of necessary, but certainly not sufficient, conditions:

• Each level set of $f$ must be a smooth submanifold, because every orbit of a smooth Lie group action is an (immersed) submanifold.
• Each compact level set of $f$ must have nonnegative Euler characteristic, because the group action restricts to a transitive action on each level set, and a theorem of Mostow shows that a compact homogeneous space must have nonnegative Euler characteristic.