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

On the frame bundle, the torsion form equals the exterior derivative of the solder form plus the connection form acting on it.

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).

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 .