I am learning Cartan's moving frame from chapter 6 of S.S. Chern's book "Lectures on differential geometry" (Google books). Suppose the moving frame in $E^N$ is denoted by $(p;e_1,\cdots,e_N)$, then we can apply an infinitesimal motion on $(p;e_1,\cdots,e_N)$ to get the equation (3.22) on p.202 Chern's book:
\begin{equation}
\left\{ \begin{split}
dp=\displaystyle{\sum_{\alpha=1}^{N}\omega^{\alpha}e_{\alpha}}\\
de_{\alpha}=\displaystyle{\sum_{\beta=1}^{N}\omega^{\beta}_{\alpha}e_{\beta}}
\end{split} \right.
\end{equation}
where the $\omega^{\alpha}$ and $\omega^{\beta}_{\alpha}$ are one forms. In my understanding, the $e_{\alpha}$'s are understood as $\mathbb{R}^N$ valued functions (of some $N$ variables $(u_1,\cdots,u_N)$) since we can do the operator $d$ to them. In other words, the $e_{\alpha}$'s are not really vector fields in $\mathbb{R}^N$ (by vector field I mean a contravariant 1tensor field) since we can't do the $d$ operator to a vector field. However, Chern's book treat the $e_{\alpha}$'s as vector fields and it takes the one forms $\omega^{\beta}_{\alpha}$ as connection 1form as the LeviCivita connection for $\mathbb{R}^N$ and do covariant derivative $D$ to $e_{\alpha}$ (this is (3.38) on p.207 Chern's book):
$$
De_{\alpha}=\omega_{\alpha}^{\beta}e_{\beta}
$$
(Here, I take $m=N$ in Chern's book). So my question is why we can treat the $e_{\alpha}$ as contravariant fileds in $\mathbb{R}^N$ and do the $d$ operator to them? Why the one forms $\omega_{\alpha}^{\beta}$ (gotten from the equation for the moving frame) are exactly the connection 1forms for the LeviCivita connection on $\mathbb{R}^N$?

4$\begingroup$ You might consider reading the recent book of Jeanne Clelland, From Frenet to Cartan: The Method of Moving Frames. It has more information on this question and is easier to read and more detailed. $\endgroup$ – Ben McKay Jul 15 '19 at 16:42
One way to make sense of this is to view $p, e_1, \dots, e_N$ as functions on the orthonormal frame bundle of $\mathbb{R}^N$, which is naturally isomorphic to the group of rigid motions, where there is a right action of the group $O(N)$ of rotations, which fixes the point $p$ and rotates the frame and $\mathbb R^N$ is the set of left cosets with respect to this action. The $1$forms are now well defined on the frame bundle and are in fact the left invariant $1$forms (also known as the MaurerCartan forms) on the group. The equations satisfied by these 1forms are the MaurerCartan equations for the group of rigid motions.
If you're familiar with Lie groups and homogeneous spaces, a nice exposition of this is in a paper of Griffiths: On Cartan's method of Lie groups and moving frames as applied to uniqueness and existence questions in differential geometry, Duke Math. J. 41 (1974), 775–814.

1$\begingroup$ I have read Griffiths' paper. I think the point is that we treat $\mathbb{R}^N$ as the quotient space of the Lie group $E(N)$ (which is isomorphic to the frame bundle) under the action of $O(N)$ so that the ambient space (i.e. $\mathbb{R}^N$) of the functions $\psi:\,E(N)\rightarrow\mathbb{R}^N$ (p 783 of Griffiths paper) can be contanined in $E(N)$ so that the equation $d\psi(F)=\displaystyle{\sum_{\alpha=1}^{N}}\psi_{\alpha}(F)e_{\alpha}$ makes sense. Am I right? $\endgroup$ – J.Doe Jul 15 '19 at 22:00

$\begingroup$ Can you expain how to show that the forms $\psi_{\alpha}$ in $d\psi(F)=\displaystyle{\sum_{\alpha=1}^{N}}\psi_{\alpha}(F)e_{\alpha}$ are left invariant on $E(N)$ on p.783 of Griffiths's paper ? and can you recommand some reference for Lie groups and homogeneous space and the principal bundles (concerning this problem)? $\endgroup$ – J.Doe Jul 15 '19 at 22:18

1$\begingroup$ @JDoe, I think Ben McKay's recommendation of Jeanne Clelland's book is a good one. $\endgroup$ – Deane Yang Jul 16 '19 at 2:33