Group action map of a Lie group on a smooth manifold being submersion

Question

Let $G$ be a Lie group action on a manifold $M$.

It is assumed that the action is smooth i.e., $G\times M\rightarrow M$ given by $(g,m)\mapsto gm$ is a smooth map.

I am trying to understand in what cases this action map is a submersion.

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I am looking at this to see that Translation groupoid $G\ltimes M$ is actually a groupoid.

We have source map $s:(G\ltimes M)_1\rightarrow (G\ltimes M)_0$ is just the map $s:G\times M\rightarrow M$ , projection to second term $(g,m)\mapsto m$. This is a submersion.

We have target map $t:(G\ltimes M)_1\rightarrow (G\ltimes M)_0$ is the action map $t:G\times M\rightarrow M$, $(g,m)\mapsto gm$. I can not see why this has to be a submersion in general.

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If you are not familiar with the notion of Lie groupoid, that is fine, you can atleast help me to see when the action map is a submersion.

Any suggestions are welcome.

Answer 1

Always. The differential is onto, since the map $m \in M \mapsto g_0 m \in M$ is a diffeomorphism, for all $g_0 \in G$.