I want to grasp the moving frames method but I find some obstacles. I don't know if this question is suitable for MO, if it is not the case please let me know and I will move it.  

I am aware there are other related questions here like [this one](https://mathoverflow.net/questions/337294/moving-frames-method-for-non-matrix-lie-group) or [this one](https://mathoverflow.net/questions/336149/about-the-cartans-moving-frame-method?rq=1), but they don't answer my doubts.


Given a Lie group $G$ and a homogeneous space $X\equiv G/H$, the goal of the moving frames method is to study submanifolds $M$ of $X$. In particular we want to know if two given submanifolds $M$ and $\tilde{M}$ are "congruent", in the sense that there is a "movement" $g\in G$ such that $g(M)=\tilde{M}$.


I know I have a $\mathfrak{g}$-valued differential form on $G$ which is left invariant, the Maurer-Cartan form $\theta$. The left-invariance of $\theta$ allows us to show (see [griffiths1974cartan](https://www.google.com/url?sa=t&rct=j&q=&esrc=s&source=web&cd=&cad=rja&uact=8&ved=2ahUKEwjxmv-g3tf6AhUCcxoKHQC6DLcQFnoECAsQAQ&url=https%3A%2F%2Fprojecteuclid.org%2Fjournals%2Fduke-mathematical-journal%2Fvolume-41%2Fissue-4%2FOn-Cartans-method-of-Lie-groups-and-moving-frames-as%2F10.1215%2FS0012-7094-74-04180-5.full&usg=AOvVaw3bbvtFV6C_skrb20r37Tg9), lemma (1.3)) that given two maps $f,\tilde{f}$ from, let's say, $\mathbb{R}^n$ to $G$ then $\tilde{f}(x)=g\cdot f(x)$ if and only if $\tilde{f}^\*(\theta)=f^\*(\theta)$. This way $f^\*(\theta)$ provide a set of invariants to characterize submanifolds of $G$ (not of $X$!).


Suppose our submanifolds $M$ and $\tilde{M}$ are parametrized respectively by maps $\alpha:\mathbb{R}^n \to X$ and $\tilde{\alpha}:\mathbb{R}^n \to X$. If we are in a particular case, for example $G=E(2)$, $X=\mathbb{E}^2$ and $M,\tilde{M}$ curves, we have a "canonical" way to lift $\alpha$ and $\tilde{\alpha}$ to $f\_{\alpha}, f\_{\tilde{\alpha}}:\mathbb{R} \to G$ (the unitary tangent vector and its orthogonal, together with the curve point itself). This way, if
$$
f\_{\alpha}^\*(\theta)={f}\_{\tilde{\alpha}}^\*(\theta)
$$
we conclude that the "curves of frames" $f\_{\alpha}$ and $f\_{\tilde{\alpha}}$ are congruent, and therefore $\alpha$ and $\tilde{\alpha}$ are congruent.


The key fact here is, I think, that the assignment
$$
\alpha \mapsto f\_{\alpha}
$$
is $G$-invariant, in the sense that $\tilde{\alpha}=g\alpha$ if and only if $f\_{\tilde{\alpha}}=g f\_{\alpha}$. Otherwise we could have congruent curves in $E(2)$ which couldn't be detected by the invariants (because of the "bad assignment" of frames to the curves).


**Question 1**  

Back to the general case: can we always find such a "canonical lift"? Is there a method to find it? Or is the moving frames method restricted to a bunch of particular cases?  

$\blacksquare$


**Question 2**  

Can you provide at least a brief list of examples of these assignments? For example:


* Curves in $\mathbb{R}^3$ with Euclidean movements: the Frenet frame.
* Surfaces in $\mathbb{R}^3$ with Euclidean movements: a frame made with the point, the normal vector and two ortogonal vectors aligned with the principal directions of the surface (or is not this necessary?).
* ...


$\blacksquare$


I have read the article of Griffiths, the corresponding chapter of *Cartan for begginers* and *From Frenet to Cartan*, but I am still blocked with these doubts.



Moving frames method
I think you might want to read a couple of articles on the moving frame that carefully discuss this issue (and show that it is more subtle than most people realize).


The first is a paper by Mark Green, *The moving frame, differential invariants and rigidity theorems for curves in homogeneous spaces* (Duke Math. J. 45 (1978), no. 4, 735–779). Mark discusses the 1-dimensional case in great detail and shows, by example, that, in fact, there are cases in which a `canonical' lifting by Cartan's method does not exist, and one must extend the method by introducing a family of liftings in order to find all of the invariants.


The second is a set of lecture notes by Gary Jensen,
*Higher order contact of submanifolds of homogeneous spaces*,
(Lecture Notes in Mathematics, Vol. 610. Springer-Verlag, Berlin-New York, 1977.) He discusses the foundations of the subject in great detail and gives a number of illuminating examples.


Of course, there are lots of more recent papers and articles, but I think that these two illustrate a lot of the issues and point out some of the pitfalls that aren't apparent at first sight.


