Torsion 2-form

The R^n-valued 2-form on a frame bundle that measures failure of a connection to be torsion-free.

Torsion 2-form

Let $M$ be an $n$ -dimensional smooth manifold , let $FM\to M$ be its frame bundle, and let $\theta$ be the solder form .

Fix a principal connection on $FM$ , with connection 1-form $\omega$ valued in $\mathfrak{gl}(n,\mathbb{R})$ .

Definition (Torsion 2-form)

The torsion 2-form of $\omega$ is the $\mathbb{R}^n$ -valued 2-form

\Theta \in \Omega^2(FM;\mathbb{R}^n)

defined by the (first) Cartan structure equation

\Theta := d\theta + \omega \wedge \theta.

Here $\omega\wedge\theta$ denotes the natural action of $\mathfrak{gl}(n,\mathbb{R})$ on $\mathbb{R}^n$ combined with the wedge product of differential forms.

This torsion form corresponds on the base to the torsion tensor of the induced connection $\nabla$ on $TM$ :

T(X,Y) := \nabla_XY - \nabla_YX - [X,Y],

for vector fields $X,Y$ .

A connection is torsion-free if and only if $\Theta=0$ (equivalently, $T\equiv 0$ ). The Levi–Civita connection is characterized by torsion-free plus metric compatibility (see Levi–Civita connection ).

Examples

Levi–Civita. For any Riemannian manifold, the torsion 2-form of the Levi–Civita connection vanishes identically.
Left-invariant “zero” connection on a Lie group. On a Lie group $G$ with a chosen left-invariant framing, declaring the frame to be parallel (so connection coefficients vanish in that frame) produces torsion equal to minus the bracket of left-invariant fields, hence typically nonzero.
A coordinate connection with asymmetric Christoffel symbols. On $\mathbb{R}^n$ , defining a connection by $\nabla_{\partial_i}\partial_j = \Gamma^k_{ij}\partial_k$ with $\Gamma^k_{ij}\neq \Gamma^k_{ji}$ yields torsion components $T^k_{ij}=\Gamma^k_{ij}-\Gamma^k_{ji}$ .