Cartan's second structure equation (curvature) in the frame bundle

Let $M$ be an $n$-dimensional smooth manifold and let $\pi:F(M)\to M$ be its frame bundle, a principal G-bundle with structure group $G=\mathrm{GL}(n,\mathbb{R})$. Fix a principal connection $\omega\in\Omega^1(F(M);\mathfrak{gl}(n,\mathbb{R}))$, where $\mathfrak{gl}(n,\mathbb{R})$ is the Lie algebra of $G$.

Statement (frame bundle formulation)

The curvature of $\omega$ is the $\mathfrak{gl}(n,\mathbb{R})$-valued $2$-form $\Omega\in\Omega^2(F(M);\mathfrak{gl}(n,\mathbb{R}))$ defined by

This identity is Cartan’s second structure equation, where $d$ is the exterior derivative and the bracket wedge is defined by

using the Lie bracket on $\mathfrak{gl}(n,\mathbb{R})$.

The form $\Omega$ is horizontal and $G$-equivariant. Under the usual correspondence between principal connections on $F(M)$ and linear connections on $TM$, $\Omega$ encodes the Riemann curvature tensor of the induced connection on the tangent bundle .

Examples

1. Flat connection in a global frame.
   If $M=\mathbb{R}^n$ with its standard frame and $\omega=0$, then $\Omega=0$ by the second structure equation.

2. Constant sectional curvature model.
   For the round sphere with its Levi-Civita connection, in a local orthonormal coframe one has curvature $2$-forms of the schematic shape

   $$\Omega^i{}_j = K\,\theta^i\wedge\theta^j,$$

   reflecting constant sectional curvature $K>0$.

3. Product connections.
   If $M=M_1\times M_2$ and the connection on $TM\cong TM_1\oplus TM_2$ is the product of connections on the factors, then $\Omega$ is block-diagonal with blocks given by the curvature forms from each factor, and there are no mixed curvature terms.