Cartan's first structure equation (torsion) in the frame bundle

Let $M$ be an $n$ -dimensional smooth manifold and let $\pi:F(M)\to M$ denote its (linear) frame bundle, viewed as a principal G-bundle with structure group $G=\mathrm{GL}(n,\mathbb{R})$ . Write a point $u\in F(M)$ over $x=\pi(u)$ as a linear isomorphism $u:\mathbb{R}^n\to T_xM$ , where $T_xM$ is the fiber of the tangent bundle.

Statement (frame bundle formulation)

There is a canonical $\mathbb{R}^n$ -valued $1$ -form (the solder form) $\theta\in\Omega^1(F(M);\mathbb{R}^n)$ defined by

\theta_u(V)\;=\;u^{-1}\bigl(d\pi_u(V)\bigr),\qquad u\in F(M),\;V\in T_uF(M).

Given a principal connection $\omega\in\Omega^1(F(M);\mathfrak{gl}(n,\mathbb{R}))$ , define the torsion $2$ -form $\Theta\in\Omega^2(F(M);\mathbb{R}^n)$ by the covariant exterior derivative of $\theta$ :

\Theta \;:=\; D\theta.

Then Cartan’s first structure equation is the identity

\boxed{\;\Theta \;=\; d\theta \;+\; \omega\wedge \theta\;}

where the $\mathfrak{gl}(n,\mathbb{R})$ -action on $\mathbb{R}^n$ is used to define $\omega\wedge\theta$ :

(\omega\wedge\theta)(V,W)\;=\;\omega(V)\cdot\theta(W)\;-\;\omega(W)\cdot\theta(V).

Remarks

The form $\Theta$ is horizontal and $G$ -equivariant; it corresponds to the torsion tensor of the induced linear connection $\nabla$ on $TM$ , namely

T_\nabla(X,Y)=\nabla_XY-\nabla_YX-[X,Y],

where $[X,Y]$ is the Lie bracket of vector fields. In particular, $\nabla$ is torsion-free if and only if $\Theta=0$ , equivalently

d\theta+\omega\wedge\theta=0.

Examples

Euclidean space with the standard flat connection. On $M=\mathbb{R}^n$ with its global coordinate frame, the induced connection has $\omega=0$ in that frame and $\theta$ pulls back to the standard coframe. Hence $d\theta=0$ and $\Theta=0$ .

Levi-Civita connection (torsion-free case). For a Riemannian manifold, the Levi-Civita connection is torsion-free, so in any local orthonormal coframe $\{\theta^i\}$ with connection $1$ -forms $\{\omega^i{}_j\}$ one has the classical component form $d\theta^i+\omega^i{}_j\wedge\theta^j=0,$ which is exactly $\Theta=0$ expressed using the first structure equation.

Teleparallel (Weitzenböck) connection on a Lie group. Let $M=G$ be a Lie group with a global left-invariant frame. The connection for which that frame is parallel has $\omega=0$ in that trivialization, but typically $d\theta\neq 0$ for the left-invariant coframe. Then $\Theta=d\theta$ encodes the structure constants of the corresponding Lie algebra.