of the special orthogonal Lie algebra and the abelian translation Lie algebra. Accordingly, a connection 1-form has two components

$\Omega \in \Omega^1(U,\mathfrak{so}(p,q))$ (sometimes called the “spin connection”);

$E \in \Omega^1(U,\mathbb{R}^{p+q})$ (sometimes called the “vielbein”).

The metric itself is

$g = \langle E \otimes E \rangle
\,.$

Accordingly also the curvature 2-form has two components:

$R = d \Omega + [\Omega \wedge \Omega] \in \Omega^2(U, \mathfrak{so}(p,q))$ – the Riemann curvature;

$\tau = d E + [\Omega \wedge E]$ – the torsion.

Generalizations

In supergeometry a metric structure is given by a connection with values in the super Poincaré Lie algebra. The corresponding notion of torsion has an extra contribution from spinor fields: the super torsion?.