nLab
torsion of a metric connection

Context

Riemannian geometry

Differential geometry

differential geometry

synthetic differential geometry

Axiomatics

Models

Concepts

Theorems

Applications

For other notions of torsion see there.


Contents

Definition

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

This Lie algebra is the semidirect product

𝔦𝔰𝔬(p,q)𝔰𝔬(p,q) 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

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

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

The metric itself is

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

Accordingly also the curvature 2-form has two components:

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

  • τ=dE+[ΩE]\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

gauge groupstabilizer subgrouplocal model spacelocal geometryglobal geometrydifferential cohomologyfirst order formulation of gravity
generalLie group/algebraic group GGsubgroup (monomorphism) HGH \hookrightarrow Gquotient (“coset space”) G/HG/HKlein geometryCartan geometryCartan connection
examplesEuclidean group Iso(d)Iso(d)rotation group SO(d)SO(d)Cartesian space d\mathbb{R}^dEuclidean geometryRiemannian geometryaffine connectionEuclidean gravity
Poincaré group Iso(d1,1)Iso(d-1,1)Lorentz groupMinkowski space d1,1\mathbb{R}^{d-1,1}Lorentzian geometrypseudo-Riemannian geometryspin connectionEinstein gravity
super Poincaré groupspin groupsuper Minkowski spacetime d1,1|N\mathbb{R}^{d-1,1\vert N}Lorentzian supergeometrysupergeometrysuperconnectionsupergravity
linear algebraic groupparabolic subgroup/Borel subgroupflag varietyparabolic geometry
orthochronous Lorentz groupconformal geometryconformal connectionconformal gravity
generalsmooth 2-group GG2-monomorphism HGH \to Ghomotopy quotient G//HG//HKlein 2-geometryCartan 2-geometry
cohesive ∞-group∞-monomorphism (i.e. any homomorphism) HGH \to Ghomotopy quotient G//HG//H of ∞-actionhigher Klein geometryhigher Cartan geometryhigher Cartan connection
examplesextended super Minkowski spacetimeextended supergeometryhigher supergravity: type II, heterotic, 11d
Revised on December 18, 2014 14:44:39 by Urs Schreiber (127.0.0.1)