Let $G$ be a Lie group and $M$ a manifold such that $G$ acts on $M$. Let $G_a$ denote the stabilizer of an element $a \in M$.
We can define a map $f: G/G_a \to M$ which sends the coset $gG_a$ to $g\cdot a \in M$, and this map is well-defined. I also need to show that it is smooth, but I'm not sure how to do this.
Ordinarily I would choose charts and write the map in coordinates. However, I don't know what a chart on $G/G_a$ looks like. All I know is that the smooth structure on $G/G_a$ is such that the quotient map $p:G \to G/G_a$ is a submersion. I also know that $f$ is related to $p$ by the formula $f\circ p=\tilde{f}$, where $\tilde{f}:G \to M$ is $\tilde{f}(g)=g\cdot a$. I know $p$ and $\tilde{f}$ are smooth, but this doesn't imply $f$ is smooth as well.