Consider a Riemannian manifold $(M,g)$. The Levi-Civita connection is described, in a local frame $\{e_1,\ldots,e_n\}$, by the connection forms $\Theta$. This connection is a connection on a fiber bundle , so it has a curvature which, expressed in the local frame, turns out to be
$$ \Omega:=d \Theta-\Theta \wedge \Theta $$i.e., the curvature 2-forms.
On the other hand, the metric induces the Riemann curvature tensor $R$.
Theorem (Cartan's second structural equation). It is satisfied that
$$ R(X,Y;e_i) = \Omega_i^j(X,Y)e_j. $$$\blacksquare$
Proof:
According to the definition of the Riemann curvature tensor
$$R(X,Y;e_i)=\nabla_X\nabla_Y e_i-\nabla_Y\nabla_X e_i-\nabla_{[X,Y]}e_i.$$
and denoting by $\Theta$ the connection 1-forms, we have
$$ R(X,Y;e_i) =\nabla_X\left[\Theta^j_i(Y)e_j\right]-\nabla_Y\left[\Theta^j_i(X)e_j\right]-\Theta^j_i([X,Y])e_j= $$ $$ =X(\Theta^j_i(Y))e_j+\Theta^j_i(Y)\nabla_X e_j-Y(\Theta^j_i(X))e_j-\Theta^j_i(X)\nabla_Y e_j-\Theta^j_i([X,Y])e_j. $$Using infinitesimal Stokes' theorem we have:
$$ R(X,Y;e_i)=d\Theta^j_i(X,Y)e_j+\Theta^j_i(Y)\nabla_X e_j-\Theta^j_i(X)\nabla_Y e_j $$And then
$$ R(X,Y;e_i)=d\Theta^j_i(X,Y)e_j+\Theta^j_i(Y)\Theta^k_j(X)e_k-\Theta^j_i(X)\Theta^k_j(Y)e_k= $$and renaming indices
$$ R(X,Y;e_i)=d\Theta^j_i(X,Y)e_j-\Theta^k_i \wedge \Theta_k^j (X,Y) e_j= $$ $$ =\left(d\Theta^j_i(X,Y)-\Theta^k_i \wedge \Theta_k^j (X,Y) \right)e_j $$And by definition of curvature 2-forms
$$ R(X,Y;e_i) = \Omega_i^j(X,Y)e_j. $$1. In a flat space $d \Theta=\Theta \wedge \Theta$.
2. Related to the Maurer-Cartan equation, see Generalization of the flatness of R3.
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Author of the notes: Antonio J. Pan-Collantes
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