I am learning Cartan's moving frame from chapter 6 of S.S. Chern's book "Lectures on differential geometry" ([Google books](https://books.google.ca/books?id=Mvk7DQAAQBAJ&printsec=frontcover&hl=zh-CN#v=onepage&q&f=false)). 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 $\mathbb{R}^N$?



About the Cartan's moving frame method
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. 


