I am coming from [this question](https://mathoverflow.net/questions/336149/about-the-cartans-moving-frame-method?noredirect=1&lq=1), which has not being completely answered but I think is very interesting.


In several works ([Chern], [Griffiths] and [Clelland]) the Maurer-Cartan form for $E(n)$ is worked out in the following manner. They consider maps from $G=E(n)$ to $\mathbb{R}^n$, $x, e\_1,\ldots, e\_n$, and express their differentials in terms of the frame in which we are. But for me that doesn't seem natural because is something very particular of this example: the frame itself can be described in terms of the objects it describe. I consider more natural the general approach: the group $E(n)$ can be seen like a matrix group of a special type, that one with elements of the form
$$
\begin{pmatrix}
A & v\\
0 & 1\\
\end{pmatrix}
$$
with $A\in O(n)$ and $v\in \mathbb{R}^n$. And now you only have to apply the formula for MC form for a matrix group, $\theta=g^{-1}dg$, obtaining the same 1-forms.



> 
> **Question 1 (solved)**  
> 
> Is this true for every Lie group of this type? That is, whenever we have a group $G\approx \mathbb{R}^n \rtimes H$ it can be seen as a subgroup of $GL(n+1)$ as above (see [this QA in MSE](https://math.stackexchange.com/questions/4554039/semidirect-product-and-matrix-groups/4554089#4554089)) and we can interpret the columns as vectors in the homogeneous space $G/H\approx \mathbb{R}^n$. Then, does the Maurer-Cartan form tell us the variation of these vectors expressed in the current frame?
> 
> 
> 


Back to the case of $E(2)$, for simplicity. The MC form is
$$
\theta=g^{-1}dg=\begin{pmatrix}
0&-d\theta&cos(\theta)da+sin(\theta)db\\
d\theta&0&-sin(\theta)da+cos(\theta)db\\
0&0&0&\\
\end{pmatrix}
$$
If we consider the basis of $\mathfrak{e}(2)$ given by
$$
B=\left\{
\begin{pmatrix}
0&0&1\\
0&0&0\\
0&0&0
\end{pmatrix},
\begin{pmatrix}
0&0&0&\\
0&0&1\\
0&0&0&\\
\end{pmatrix},
\begin{pmatrix}
0&-1&0&\\
1&0&0\\
0&0&0&\\
\end{pmatrix}
\right\}\equiv
$$
$$
\equiv\{\partial\_a|\_e,\partial\_b|\_e,\partial\_{\theta}|\_e\}
$$
we have
$$
\theta=\mu\_1 \otimes\partial\_a|\_e+\mu\_2 \otimes\partial\_b|\_e +\mu\_3\otimes \partial{\theta}|\_e
$$
with
$$
\mu\_1=cos(\theta)da+sin(\theta)db
$$
$$
\mu\_2=-sin(\theta)da+cos(\theta)db
$$
$$
\mu\_3=d\theta
$$
In this case the Maurer-Cartan form has "two parts": $\mu\_1, \mu\_2$ on the one hand, and $\mu\_3$ on the other hand. I think that $(\mu\_1, \mu\_2)$ corresponds to the canonical solder form and $\mu\_3$ is the connection form of the Levi-Civita connection.



> 
> **Question 2**  
> 
> Why is this the Levi-Civita connection? What relationship does it have (if any) with the group reduction of $GL(2)$ to $O(2)$ by means of the standard metric?
> 
> 
> 


I have an intuition about some relation but I cannot grasp what it is... I know that the frame bundle for $\mathbb{R}^2$ is $\mathbb{R} \rtimes GL(2)$ and that the standard metric let us reduce the structure group of this principal bundle to $O(2)$...


**References**  

[Chern]: Chapter 6 of S.S. Chern's book "Lectures on differential geometry"


[Griffiths]: On Cartan's method of Lie groups and moving frames as applied to uniqueness and existence questions in differential geometry


[Clelland]: From Frenet to Cartan: The Method of Moving Frames



Maurer-Cartan form and Levi-Civita connection
I have been working in question 2 and I think I have a good explanation. At the end is a triviality, but that's what (almost) always happens in math when you understand something.


I am going to discuss here the Euclidean plane from two different perspectives.


Euclidean plane
===============


From the point of view of classical differential geometry
---------------------------------------------------------


The Euclidean plane is the manifold $M=\mathbb{R}^2$, with coordinates $(x\_1,x\_2)$, together with the natural [Riemannian metric](https://ajpancollantes.github.io/%E2%99%BE%EF%B8%8F%20CONCEPTS/GEOMETRY/Riemannian%20metric.md.html)
$$
g=dx\_1\otimes dx\_1+dx\_2\otimes dx\_2.
$$
It is, therefore, a Riemannian manifold.


Seen as a plain manifold we can consider that $M$ is endowed with a natural linear connection $\nabla$, such that $\nabla\_{\partial\_{x\_i}}\partial\_{x\_j}=0\partial\_{x\_1}+0\partial\_{x\_2}$. The [induced](https://ajpancollantes.github.io/%E2%99%BE%EF%B8%8F%20CONCEPTS/GEOMETRY/associated%20connection.md.html#Example) principal connection on the frame bundle is given by the 1-form
$$
\omega=
\begin{pmatrix}
c\_{11}&c\_{12}\\
c\_{21}&c\_{22}\\
\end{pmatrix}^{-1}\cdot\begin{pmatrix}
dc\_{11}&dc\_{12}\\
dc\_{21}&dc\_{22}\\
\end{pmatrix}\in \Omega(\mathbb R^2,\mathfrak{gl}(2))
$$
at
$$
f=
\begin{pmatrix}
c\_{11} & c\_{12}& x\_1\\
c\_{21} & c\_{22}& x\_2\\
0 & 0& 1\\
\end{pmatrix}\in FM
$$
[Here](https://ajpancollantes.github.io/%E2%99%BE%EF%B8%8F%20CONCEPTS/GEOMETRY/associated%20connection.md.html#Example) it is shown how to construct $\omega$ from an arbitrary $\nabla$.


On the other hand, if we think of $M$ as a Riemannian manifold we can consider the [Levi-Civita connection](https://ajpancollantes.github.io/%E2%99%BE%EF%B8%8F%20CONCEPTS/GEOMETRY/Levi-Civita%20connection.md.html) $\nabla\_{LC}$. Since the metric is constant, the covariant derivative $\nabla\_{LC}$ coincides with the natural covariant derivative $\nabla$, so it induces the same connection $\omega$ on $FM$.
But the metric $g$ also specifies a orthonormal frame bundle $OM$ ([see here why](https://ajpancollantes.github.io/%E2%99%BE%EF%B8%8F%20CONCEPTS/orthonormal%20frame%20bundle.md.html)). The elements of this principal bundle are
$$
f=
\begin{pmatrix}
c & -\sqrt{1-c^2}& x\_1\\
\sqrt{1-c^2} & c& x\_2\\
0 & 0& 1\\
\end{pmatrix}\in OM
$$
with $c\in [-1,1]$.
Since
$$
\omega|\_{OM}=\begin{pmatrix}
c&\sqrt{1-c^2}\\
-\sqrt{1-c^2}&c\\
\end{pmatrix}\cdot\begin{pmatrix}
dc&\frac{c}{\sqrt{1-c^2}}dc\\
\frac{-c}{\sqrt{1-c^2}}dc&dc\\
\end{pmatrix}=
$$
$$
=\begin{pmatrix}
0&\frac{dc}{\sqrt{1-c^2}}\\
-\frac{dc}{\sqrt{1-c^2}}&0\\
\end{pmatrix} \in \Omega(\mathbb R^2,\mathfrak{o}(2)),
$$
according to Proposition 4.7. in [vicenteBundles](https://www.google.com/url?sa=t&rct=j&q=&esrc=s&source=web&cd=&cad=rja&uact=8&ved=2ahUKEwj-w6a2xYj7AhVJ3hoKHVDpAMQQFnoECAsQAQ&url=http%3A%2F%2Fwww.mat.ucm.es%2F%7Evmunozve%2FNotas-fibrados.pdf&usg=AOvVaw2EmcybsLZEb16epRLqS8z8), this connection can be reduced to a connection on the orthonormal frame bundle determined by the metric $g$.


If we parameterize this principal bundle $OM$ with
$$
(x\_1,x\_2,\theta)\mapsto \begin{pmatrix}
\cos(\theta) & -\sin(\theta)& x\_1\\
\sin(\theta) & \cos(\theta)& x\_2\\
0 & 0& 1\\
\end{pmatrix}
$$
we obtain the more famous expression for $\omega$:
$$
\begin{pmatrix}
0&d\theta\\
-d\theta&0\\
\end{pmatrix}
$$
Remember: this 1-form **tells us how much the bases at $f$ and $f'$ "fail to be constant"** when we pass from the frame $f$ to a nearby frame $f'$, but expressing this mistake with respect to the frame $f$ itself.


From the point of view of Cartan geometry
-----------------------------------------


The Euclidean plane is a Cartan geometry modeled over $(E(2),O(2))$ , indeed **is** the Klein geometry $(E(2),O(2))$. Moreover, it is a [reductive Klein geometry](https://ajpancollantes.github.io/%E2%99%BE%EF%B8%8F%20CONCEPTS/reductive%20Cartan%20geometry.md.html#Interpretation) since
$$
\mathfrak e(2)=\left\{\begin{pmatrix}
C & v\\
0 & 0\\
\end{pmatrix} 
:C\in \mathfrak{o}(2), v\in \mathbb R^2\right\}=\mathfrak o(2)\oplus \mathfrak p
$$
We have a natural choice for $\mathfrak p$
$$
\mathfrak{p}=\left\{\begin{pmatrix}
0 & p\\
0 & 0\\
\end{pmatrix} 
:p\in \mathbb R^2\right\}.
$$


With this in mind, remember that the Maurer-Cartan form describes all possible "infinitesimal displacements" of the frame $f$, but from the point of view of the frame $f$ itself. That is, if we pass from the frame $f$ to another frame $f'$, the Maurer-Cartan form at $f$ applied to the "vector" $\vec{ff'}=(dx\_1,dx\_2,d\theta)$ is a packet of information
$$
A=\begin{pmatrix}
0&-d\theta&cos(\theta)dx\_1+sin(\theta)dx\_2\\
d\theta&0&-sin(\theta)dx\_1+cos(\theta)dx\_2\\
0&0&0&\\
\end{pmatrix}\in \mathfrak e(2)
$$
Here is encoded, on the one hand, how much have we moved the base point of $f$ to the base point of $f'$ and, on the other, how much have we changed the basis itself. The natural choice of $\mathfrak p$ let us think that the information about the change of base point is in the $v$ part (the projection of the Maurer-Cartan form on $\mathfrak p$), and so the projection of the Maurer-Cartan form on $\mathfrak o(2)$, $\begin{pmatrix}0&d\theta\\-d\theta&0\\\end{pmatrix}$, tell us how much has the basis changed. That is, the same as the connection 1-form of the connection on the orthonormal bundle induced by the metric $g$ (which is the Levi-Civita connection).


**To summarize:**  

In the orthonormal frame bundle induced by the metric $g$ we consider a displacement from a frame $f=\begin{pmatrix}C & p\\0 & 1\\\end{pmatrix}$ to a frame $f'=\begin{pmatrix}C' & p'\\0 & 1\\\end{pmatrix}$.


The principal connection $\omega$ induced by the Levi-Civita connection measures the change from $C$ to $C'$ as an element of $\mathfrak o(2)$.


The Cartan connection (Maurer-Cartan form) measures the change from $f$ to $f'$ as an element of $\mathfrak e(2)$. This change can be decomposed like the union of a change from $C$ to $C'$ and a change from $p$ to $p'$. This is reflected in the fact that $\mathfrak e(2)=\mathfrak o(2)\oplus \mathfrak p$. If we focus on the change from $C$ to $C'$ we have the principal connection $\omega$.


