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!


1 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.

  • $\begingroup$ Thank you very much! I've read that the theorem you mention is a consequence of the E. Cartan's "Equivalence Theorem of local coframes" [see Theorem 8.15 of Olver, P. (1995). Equivalence, Invariants and Symmetry. Cambridge: Cambridge University Press] In the theorem, it is implicitly assumed that "every Lie algebra g has a Lie group G whose lie algebra is isomorphic to g". I was wondering if in the Cartan's original, he constructed a local Lie group "by hand" instead of supposing it existence. $\endgroup$
    – FUUNK1000
    May 4 at 10:03
  • 1
    $\begingroup$ I don't know the details of Cartan's proof, but looking at the name, I would expect that that result you are interested in is a quite special case and the general result is not strongly related to Lie groups. Therefore, I would not expect that Cartan's proof gives a construction of the local Lie group. $\endgroup$ May 7 at 11:41

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