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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 or this one, 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, 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?).

Added:

• Curves $\alpha$ in equi-affine space $\mathbb{A}^3\equiv \mathbb{R}^3 \rtimes SL(3)/SL(3)$: an unimodular frame built from $\alpha'(t)$, $\alpha''(t)$ and $\alpha'''(t)$. I find it a particularly iluminating example. Is very well explained in Clelland's book, From Frenet to Cartan, page 172.
  $\blacksquare$

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.

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 or this one, 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, 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.

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 or this one, 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, 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.

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 or this one, 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, 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?).

Added:

• Curves $\alpha$ in equi-affine space $\mathbb{A}^3\equiv \mathbb{R}^3 \rtimes SL(3)/SL(3)$: an unimodular frame built from $\alpha'(t)$, $\alpha''(t)$ and $\alpha'''(t)$. I find it a particularly iluminating example. Is very well explained in Clelland's book, From Frenet to Cartan, page 172.
  $\blacksquare$

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.

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 or this one, 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, 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 assignation $$ \alpha \mapsto f_{\alpha} $$ 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 assignations" 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 assignations? 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.

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 or this one, 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, 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.