Let $G$ be a Lie group and let $H$ be a Lie subgroup of $G$. Let $M$ be a smooth manifold on which $H$ acts from the left.
Let's consider the action of $H$ on $G \times M$ : $$h((g,m)):= (gh,h^{-1}m), \quad h \in H , g \in G , m \in M, $$ and define the manifold $Z$ to be the quotient $G \times_H M .$
I'm reading some lecture notes and I'm stuck to understand the following statement in it:
The tangent space $ T_{[g,m]} Z$ is identified with $T_gG \times T_m M/\sim, $ where $\sim$ is the relation of equivalence coming from the $H$ -orbits in $G \times M.$
Can someone please explain this identification.
Ps. My thoughts about this was to use the fact that $G \times_H M \rightarrow G/H$ is a fiber bundle and use local trivialization to identify $T_{[g,m]}Z$ with $T_mM \times T_{[g]}(G/H)$, but I don't know how to obtain the identification in the lecture notes.