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 1-tensor 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 1-form as the Levi-Civita 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 1-forms for the Levi-Civita connection on $\mathbb{R}^N$?

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    $\begingroup$ 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 Maurer-Cartan forms) on the group. The equations satisfied by these 1-forms are the Maurer-Cartan 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.

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    $\begingroup$ 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
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    $\begingroup$ @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

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