nLab
torsion of a metric connection

For other notions of torsion see there.

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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 $\mathrm{𝔦𝔰𝔬}(p,q)$.

This Lie algebra is the semidirect product

$$\mathrm{𝔦𝔰𝔬}(p,q)\simeq \mathrm{𝔰𝔬}(p,q)⋉{ℝ}^{p+q}$$\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

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

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

The metric itself is

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

Accordingly also the curvature 2-form has two components:

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

• $\tau =dE+[\Omega \wedge E]$ – 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?.

Related concepts

• first-order formulation of gravity