# Moving frame method with non-matrix Lie group

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?

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 $$g\in G$$.
• 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. – Ted Shifrin Dec 3 '18 at 23:47