Reconstructing a Lie group from its Maurer-Cartan form (role of completeness)

Question

Theorem III.8.7 in Sharpe and Chern's "Differential Geometry: Cartan's Generalization of Klein's Erlangen Program" states:

If $M$ is a simply connected manifold, $\mathfrak{g}$ is a Lie algebra, and $\omega\in\Omega^1(M,\mathfrak{g})$ is such that

1. $\mathrm{d} \omega+\frac12[\omega,\omega]=0$;
2. $\omega_x:T_xM\to \mathfrak{g}$ is an isomorphism for each $x\in M$;
3. $\omega$ is complete (i.e. each constant vector field is complete),

then $M$, with any choice of identity, has a unique Lie group structure whose Maurer-Cartan form is $\omega$.

My issue is with the proof: it never explicitly uses $\omega$'s completeness. In fact, it cites a "Fundamental Theorem" that doesn't directly apply:

Fundamental Theorem:

If $(M,x_0)$ is simply connected, $G$ is a Lie group with Maurer-Cartan form $\theta_G$, and $\eta\in \Omega^1(M,\mathfrak{g})$ satisfies the structure equation $\mathrm{d}\eta+\frac12[\eta,\eta]=0$, then for any $g\in G$ there is a unique smooth map $f:M\to G$ such that $f(x_0)=g$ and its logarithmic derivative $\delta f:=f^\ast\theta_G$ coincides with $\eta$.

The idea is to construct maps $m:M\times M\to M$ and $i:M\to M$ that act as the multiplication and inversion map. For that, they construct $1$-forms $\eta$ that are expressed in terms of $\omega$ the same way log-derivatives of a multiplication or an inversion map would be in terms of the MC form, and then check that $\mathrm{d}\eta+\frac12[\eta,\eta]=0$. Then they cite the Fundamental Theorem to reconstruct two unique maps $m:M\times M\to M$ and $i:M\to M$. However, the Fundamental Theorem obviously doesn't apply because the target manifold, $M$, isn't a Lie group yet.

This seems to be the place where completeness should come in. I suppose one needs to show that the Fundamental Theorem holds not just for Lie group targets, but for all targets with a $1$-form $\omega$ that satisfies all 3 conditions above. The book skips over this part, proving that the Maurer-Cartan form can be developed along any path before they even discuss the concept of completeness.

Does anyone have intuition for what is necessary to complete the proof?

Answer 1

On $M\times G$, the Pfaffian system $\omega-g^{-1}dg$ arises from a foliation, by the Frobenius theorem. Let $\tilde{M}$ be a leaf, i.e. a maximal integral manifold of the foliation, through some point $(m_0,1)\in M\times G$. The constant vector fields then live on $\tilde{M}$, pulled back from both local diffeomorphisms to $M$ and to $G$, since locally $\tilde{M}$ is the graph of a local diffeomorphism matching the Maurer-Cartan forms, so matching the constant vector fields. By a theorem of Ehressman (Theorem B.2 in my Introduction to Cartan geometries, but the proof is very easy), the completeness of the vector fields ensures that both $\tilde{M}\to M$ and $\tilde{M}\to G$ are covering maps. This is where the completeness is needed: a local diffeomorphism matching up complete vector fields on connected manifolds is a covering map.

Every connected covering space of a connected Lie group is a Lie group for a unique Lie group structure for which the covering map is a local isomorphism of Lie groups. I think you can find the proof of that statement in many textbooks on Lie groups. The covering map $\tilde{M}\to G$ makes $\tilde{M}$ a Lie group. The covering map $\tilde{M}\to M$ is a diffeomorphism, because $M$ is connected and simply connected.