Maurer-Cartan form on a matrix Lie group

In case $G$ is a matrix Lie group, i.e., $G \subset G L (n)$ then the (left-invariant) Maurer-Cartan form can be computed with the following formula

θ = g^{- 1} d g

This is the expression appearing everywhere, but I think the correct expression for $θ$ should be, for $p \in G$ and $V \in T_{p} G$

θ_{p} (V) = d g_{e}^{- 1} (g (p)^{- 1} d g_{p} (V))

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But what is $d g$ ? Following @ivey2016cartan definition 1.6.1, $g : G \to M_{n \times n}$ is a parametrization (embedding) of the group as a submanifold of $M_{n \times n}$ but the name $g$ is a bit misleading. For example, consider for $S \subset R^{2}$ the embedding

ι : S \to R^{3}

given by $ι (a, b) = (a^{2} + b, - b, a + b)$ . The differential $d ι$ is a map

d ι : T S \to T R^{3}

and we know that for a vector $(v_{1}, v_{2}) \in T_{p} S$ the image is computed as

d ι_{p} (v) = (\begin{matrix} 2 a & 1 \\ 0 & - 1 \\ 1 & 1 \end{matrix}) \cdot (\begin{matrix} v_{1} \\ v_{2} \end{matrix}) = (\begin{matrix} 2 a v_{1} + v_{2} \\ - v_{2} \\ v_{1} + v_{2} \end{matrix}) .

If we use the natural identification $T_{ι (p)} R^{3} \approx R^{3}$ we can think of $d ι$ as the $R^{3}$ -valued differential 1-form

d ι = (2 a d a + d b, - d b, d a + d b)

In the case of $g : G \to M_{n \times n}$ , think in $G$ as the parameter space and $M_{n \times n}$ a fancy way of writing $R^{n^{2}}$ . So $d g$ is a $M_{n \times n}$ -valued differential form in $G$ . This way, the expression $θ = g^{- 1} d g$ is also a $M_{n \times n}$ -valued differential form in $G$ .

Examples
I have worked examples in xournal 188 (affine group) and xournal 189 (Euclidean group).

Interpretation

In xournal 189 there is a reflection on how do we understand the Maurer-Cartan form when the group is interpreted as a set of frames. If $G$ is the set of frames or $G$ -descriptions of a homogeneous space $X$ (see this), then for a frame $f \in G$ , a vector $V \in T_{f} G$ can be thought as an arrow connecting the frame $f$ with a "nearby" frame $\bar{f}$ , i.e., something like $V \approx \bar{f} - f \approx δ f$ . That is, $V$ is a infinitesimal change of the frame $p$ . In this context, $θ_{f} (V)$ is nothing but how this infinitesimal change can be described from the own frame $f$ .

The same in other words (following ideas of the end of this): given a frame $g \in G$ , we can consider a nearby frame $\bar{g} \approx g + d g$ . To express this "frame" in the frame $g$ we take

g^{- 1} (g + d g) = I + g^{- 1} d g .

So the Maurer-Cartan form $g^{- 1} d g$ is the "infinitesimal displacement" of the frame, but described from the point of view of the chosen frame.