# Maurer-Cartan form and Levi-Civita connection

I am coming from this question, 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) 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

In question 1, I think you are just asking, if we have elements of $$G$$ written as $$g=\begin{pmatrix}h&v\\ 0&1\end{pmatrix}$$ then $$g^{-1}=\begin{pmatrix}h^{-1}&-h^{-1}v\\ 0&1\end{pmatrix}$$ hence $$g^{-1}dg=\begin{pmatrix}h^{-1}dh&-h^{-1}dv\\ 0&1\end{pmatrix}.$$ This is clearly since the Lie group operation is matrix multiplication.

Question 2: In books which discuss the moving frame, one learns that the soldering forms $$\omega_1,\omega_2$$ are precisely those for which the projection of orthonormal frame bundle to surface pulls back the dual of the orthonormal frame to $$\omega_1,\omega_2$$ and the Levi-Civita connection form $$\omega_{12}$$ is precisely the one for which $$d\omega_1=-\omega_{12}\wedge\omega_2$$ and $$d\omega_2=\omega_{12}\wedge\omega_1$$, so you can check that $$\mu_3$$ is the Levi-Civita. For an example of such a book, my Introduction to Exterior Differential Systems discusses all of this in its many appendices, in particular in appendix G the structure equations are derived for Riemannian metrics on surfaces.

• Yes, you are right. I asked yesterday, but this morning I realized that question 1 was a triviality. I had no time to edit. Thanks for the confirmación. Now I am struggling with question 2. Oct 17, 2022 at 14:45
• Thank you for the reference, I'll take a look Oct 21, 2022 at 14:34
• I think LibGen links on MO should be discouraged (though opinion on that is far from universal), but I guess no-one can argue about you giving a link to your own book on LibGen! Oct 31, 2022 at 19:56

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 $$g=dx_1\otimes dx_1+dx_2\otimes dx_2.$$ 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 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 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 $$\nabla_{LC}$$. 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). 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, 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 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 $$\omega$$.