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I am trying to understand the modern formulation of the moving frame method for Lie group acting on a manifold. I know the following theorem

Let be $M$ a manifold, $G$ a Lie group and $\omega$ the Maurer-Cartan form of $G$. If $f_1, f_2: M \to G$ are two functions, than $f_1^*\omega = f_2^*\omega$ if and only if $f_1 = gf_2$ for a fixed $g \in G$.

I have always seen this theorem proved for matrix Lie group. In this case that's easy because $G$ can play directly with his Lie algebra $\frak{g}$ (for example we have $\omega_g = g^{-1}dg$). My question is:

Is this theorem true for a general Lie group? How is this proved?

Thanks in advance.

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There are various proofs in textbooks. My favorite proof is to use E. Cartan's graph trick and the Frobenius Theorem.

As you stated it, things aren't quite right. You need $M$ connected.

If $\omega^1,\dots,\omega^n$ are $n$ basis left-invariant forms (so we pick a basis for $\mathfrak g^*$ and pull back by $L_g$), consider the differential ideal generated by the $1$-forms $$\eta^i = f_1^*\omega^i-f_2^*\omega^i.$$ Conceptually, we're looking at the map $F=(f_1,f_2)\colon M\to G\times G$ and pulling back the forms $\phi^i=\pi_1^*\omega^i - \pi_2^*\omega^i$ by the product map. The differential system $\phi^1=\dots=\phi^n=0$ is completely integrable, since $$d\phi = \pi_1^*[\omega,\omega] - \pi_2^*[\omega,\omega] = [\phi,\pi_1^*\omega] + [\pi_2^*\omega,\phi] \equiv 0 \pmod\phi.$$ Indeed, integral manifolds of $\phi^i=0$ give left cosets of the diagonal subgroup $\Delta\subset G\times G$. Since $F^*\phi^i = 0$ by hypothesis, and since $M$ is connected the image of $F$ must be contained in one of those integral manifolds, which says that $f_1=gf_2$ for some $g\in G$.

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  • $\begingroup$ Thanks, as always, for your answer! Is necessary to use Frobenius theorem? I have always seen it used for the existence part. And which textbooks do you have in mind? $\endgroup$ – Marco All-in Nervo Dec 3 '18 at 23:18
  • $\begingroup$ Well, in this case you can get the integral manifolds by inspection. I was thinking of the more interesting parallel result that you construct $f\colon M\to G$ if you have a system of $1$-forms on $M$ satisfying the Maurer-Cartan equations of $G$. ... This stuff is in Warner and Spivak (volume 1). There's also a beautiful paper of Griffiths on Lie Groups and Moving Frames (Duke, 41, no. 4, pp. 775-814). Also, see Chern/Chen/Lam, pp. 198 ff. $\endgroup$ – Ted Shifrin Dec 3 '18 at 23:47
  • $\begingroup$ Yes, I got your answer and your proof is really beautiful. I was looking for a simpler proof (as in Spivak, thank you!) with instruments that I can manage with my little experience. The paper of Griffiths seems really ...wow. Thank you again, next time I will start my question with "Dear Shifrin, here my new doubts about moving frames..." $\endgroup$ – Marco All-in Nervo Dec 4 '18 at 20:20
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Here Spivak's proof from A Comprehensive Introduction to Differential Geometry, Vol. 1

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