Let $M$ be a manifold and $G$ a connected Liegroup acting smoothly on $M$. Take $x \in M$ and define by $ev_x \colon G \to M, \ g \mapsto g.x$ the evaluationmap at $x$.
Is it true, that $G$ acts transitively on the connected components, iff the differential of $d_e(ev_x) \colon \mathfrak{g} \to T_xM$ is a submersion?
Edit: Maybe it is important, that $ev_y$ is an submersion for all $y \in M$?