nLab
torsion of a metric connection
Context
Riemannian geometry
Differential geometry
For other notions of torsion see there.

Contents
Definition
A (pseudo ) Riemannian metric with metric-compatible Levi-Civita connection on a smooth manifold $X$ may be encoded by a connection with values in the Poincaré Lie algebra $\mathfrak{iso}(p,q)$ .

This Lie algebra is the semidirect product

$\mathfrak{iso}(p,q) \simeq \mathfrak{so}(p,q) \ltimes \mathbb{R}^{p+q}$

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 .

This is the special case of the more general concept of torsion of a Cartan connection .

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? .

See also

Revised on December 18, 2014 14:44:39
by

Urs Schreiber
(127.0.0.1)