# About the Cartan's moving frame method

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:
$$$$\left\{ \begin{split} dp=\displaystyle{\sum_{\alpha=1}^{N}\omega^{\alpha}e_{\alpha}}\\ de_{\alpha}=\displaystyle{\sum_{\beta=1}^{N}\omega^{\beta}_{\alpha}e_{\beta}} \end{split} \right.$$$$ where the $$\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$$?

• 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. – 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.
• 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? – J.Doe Jul 15 '19 at 22:00
• 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)? – J.Doe Jul 15 '19 at 22:18