Lie group action on a finite dimensional flat manifold

Question

Consider a finite dimensional flat Riemannian manifold $M$ quotiented by an action of a finite dimensional Lie group $G$, giving rise to the quotient $Q$.

First, assume that the action is isometric. Is this situation "equivalent" to $\mathbb{R}^n$ quotiented by the action of $SO(n)$? If yes, in which sense?

Second, under which assumptions do we get a flat $Q$? It is the case for a free action of $G$. Would it be the case for an isometric action of $G$?

Many thanks in advance.

Answer 1

Consider an action of the circle by translation on a flat torus; the quotient is not equivalent to $\mathbb{R}^n/\mathrm{SO}(n)$ in any meaningful sense that I can imagine, and if the translation direction is generic the quotient is not separated, so its flatness seems ill-defined. Note also that $\mathbb{R}^n/\mathrm{SO}(n)$ is a half-line, so more generally if the dimension of $G$ is less than the dimension of $M$ you cannot expect anything in the direction you suggest.