nLab torsion of a metric connection

Context

Riemannian geometry

Riemannian geometry

Applications

Differential geometry

differential geometry

synthetic differential geometry

Applications

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.

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?.