TITLE: Maurer-Cartan form and curvature form on a Lie Group
QUESTION [5 upvotes]: We know that, given a Lie group, we can build its Maurer-Cartan form $\omega_G=P^{-1}dP$ (I don't explain now the meaning of these symbols, which I think to be generally known). This is a left-invariant form and satisfies the Maurer-Cartan equation $d\omega_G+\omega_G\wedge\omega_G=0$.
My question is: how can we relate this form with the so-called "curvature forms" on the left-invariant metric (I use "the" since they're all equivalent)? I'll explain this question by steps:
1) By "connection form" I mean a matrix-valued 1-form on a manifold which satisfies $\nabla e_i=\omega^j_i\otimes e_j$ given a frame $\{e_j\}$. Is there a clear way to obtain the connection form from the M-C form?
2) Given the connection form $\omega$, we define the curvature form $\Theta=d\omega+\omega\wedge\omega$ (i.e. $\Theta^j_i=d\omega_i^j+\omega^j_k\wedge\omega^k_i$). It turns out that $\Theta$ is of the form $\frac{1}{2}R^i_{jkt}\theta^k\wedge\theta^t$, where $\{\theta^j\}$ is the dual coframe of $\{e_j\}$ and $R^i_{jkt}=R_{ijkt}$ is the $(4,0)$ curvature tensor
$$R(X,Y,Z,W)=g(\tilde R(Z,W)Y,X)$$ where $\tilde R$ is the well-known curvature $(3,1)$ tensor $\tilde R(Z,W)Y=\nabla_Z\nabla_W Y-\nabla_Z\nabla_W Y-\nabla_{[Z,W]}Y$.
3) Now, if a relation between the M-C form and the connection form $\omega$ exists, it must not be that they simply are the same thing, since in that case the Maurer-Cartan equation would say that $\Theta$ is $0$, and therefore so is the curvature tensor, which isn't true for all Lie groups.
But since the expression of $\Theta$ and the Maurer-Cartan equation are so similar I suspect that a (less trivial) relation must exist.

REPLY [2 votes]: A) We consider the trivial principal fiber bundle $P(M,G)$, with $P=M\times G$. Then a regular flat connection is defined on $P$ by considering as horizontal subspace on each $u=(x,a) \in P$ the tangent space of $M\times \{a\}$. We consider now the Maurer-Cartan form $ω_{G}$ οn $G$ and the projection map $p:M\times G \rightarrow G$ which induces the form $ω=p^{*}ω_{G}$ on $P$. This is the connection form for the regular flat connection of $P=M\times G$. It can be proved the curvature is zero: 
$dω=d(p^{*}ω_{G})=p^{*}(dω_{G})=p^{*}(-[ω_{G},ω_{G}])=-[p^{*}ω_{G},p^{*}ω_{G}]=-[ω,ω]$.
Now, set on the above $M=G$, so we have the trivial primary bundle $p:G\times G\rightarrow G$ with connection form to be the pullback of the Maurer-Cartan on $G$. Thus, for this connection the  curvature is zero. 
B) Let's now consider the same question by using another more general bundle: the primary fiber bundle $L$ of linear frames on $G$. A connection on that bundle is by definition a linear connection on the corresponding relative budle $E$ of $G$. We remark that the connection notation $∇(X,Y)$ refers to the relative bundle $E$ and it is the "infinitesimal analogue" of the connection structure on $E$, the latter induced by the connection structure on the principal bundle $L$. By Helgason's book, there is a 1-1 correspondence between the set of the left invariant connections on a Lie group $G$ and the set of the bilinear functions $α:g \times g \rightarrow g$, where $g$ is the Lie algebra, with $α(X,Y)=∇(X',Y')$, where $X',Y'$ are the left invariant fields corresponding to $X,Y$. We choose $α$ to be identically zero. So, $α(X,Y)=∇(X',Y')=0$. Then we have a corresponding linear connection on $G$. For this connection we have that the Christoffel symbols $Γ^{k}_{ij}=0$, since it is identically a zero connection, so for the connection forms we have $ω_{j}^{i}=\sum Γ^{i}_{kj}ω^{k}=0$, where $ω^{k}$ give the dual basis locally. So in the general formula $dω_{i}=-\sum ω_{k}^{i} \wedge ω^{k}+\frac{1}{2}\sum T_{jk}^{i}ω^{j}\wedge ω^{k}$ the first sum in the right side is zero and we get $dω^{i}=\frac{1}{2}\sum T_{jk}^{i}ω^{j}\wedge ω^{k}$. Here $T$ is the torsion tensor. But it can be proved $T_{jk}^{i}=Γ_{jk}^{i}-Γ_{κξ}^{i}-c_{jk}^{i}=-c_{jk}^{i}$, where $c$ are the structural constants of the group $G$. So we get the formula $dω^{i}=-\frac{1}{2}\sum c_{jk}^{i}ω^{j}\wedge ω^{k}$ which is the Maurer-Cartan equation for that connection and it s not related to the usual Maurer-Cartan equation for $ω_{G}$, see (A). The curvature of that connection is also zero, not because of the last formula, but because the connection form is identically zero.