A Maurer-Cartan form can be used to define a local Lie group structure

Question

I have heard that a Maurer-Cartan form can be used to define a local Lie group structure on a manifold. A such form is defined as follows:

Definition. Let $N$ be a $n$-dimensional smooth manifold and $\mathfrak g$ a Lie algebra. A smooth 1-form $\omega\in\Omega^1(N,\mathfrak g)$ is said to be a Maurer-Cartan form if $d\omega+\frac{1}{2}[\omega,\omega]=0.$

If one fix a point $x\in N$, one would expect to find a local neighbourhood aroung $x$ in $N$ that admits a Lie group structure.

Is there any classical reference where I can see the proof of this?

Thanks in advance!

Answer 1

This is only true if $\dim(N)=\dim(\mathfrak g)$. The result is based on the characterization of solutions of the Maurer-Cartan equations as left logarithmic derivatives. For a Maurer-Cartan Form $\omega\in\Omega^1(N,\mathfrak g)$ and any point $x\in N$, there exists a unique smooth map from a neighborhood $U$ of $x$ in $M$ to a nighborhood of $e$ in any Lie group $G$ with Lie algebra $\mathfrak g$ such that $\omega=f^*\omega_{MC}$, where $\omega_{MC}$ is the left Maurer Cartan form on $G$. If $N$ and $G$ have the same dimension, it follows readily that $f$ has to be a local diffeomorphism, which leads to the local Lie group structure that you are looking for.

A nice exposition of this can be found in R. Sharpe's book "Differential Geometry, Cartan's generalization of Klein's Erlangen program", Springer Graduate Texts in Mathematics 166.