# 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.

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?.

generalLie group/algebraic group $G$subgroup (monomorphism) $H \hookrightarrow G$quotient (“coset space”) $G/H$Klein geometryCartan geometryCartan connection
examplesEuclidean group $Iso(d)$rotation group $SO(d)$Cartesian space $\mathbb{R}^d$Euclidean geometryRiemannian geometryaffine connectionEuclidean gravity
Poincaré group $Iso(d-1,1)$Lorentz groupMinkowski space $\mathbb{R}^{d-1,1}$Lorentzian geometrypseudo-Riemannian geometryspin connectionEinstein gravity
super Poincaré groupspin groupsuper Minkowski spacetime $\mathbb{R}^{d-1,1\vert N}$Lorentzian supergeometrysupergeometrysuperconnectionsupergravity
generalsmooth 2-group $G$2-monomorphism $H \to G$homotopy quotient $G//H$Klein 2-geometryCartan 2-geometry
cohesive ∞-group∞-monomorphism (i.e. any homomorphism) $H \to G$homotopy quotient $G//H$ of ∞-actionhigher Klein geometryhigher Cartan geometryhigher Cartan connection