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

geometric contextgauge group stabilizer subgroup local model space local geometry global geometry differential cohomology first order formulation of gravity differential geometry Lie group /algebraic group $G$ subgroup (monomorphism ) $H \hookrightarrow G$ quotient (“coset space ”) $G/H$ Klein geometry Cartan geometry Cartan connection
examples Euclidean group $Iso(d)$ rotation group $O(d)$ Cartesian space $\mathbb{R}^d$ Euclidean geometry Riemannian geometry affine connection Euclidean gravity
Poincaré group $Iso(d-1,1)$ Lorentz group $O(d-1,1)$ Minkowski spacetime $\mathbb{R}^{d-1,1}$ Lorentzian geometry pseudo-Riemannian geometry spin connection Einstein gravity
anti de Sitter group $O(d-1,2)$ $O(d-1,1)$ anti de Sitter spacetime $AdS^d$ AdS gravity
de Sitter group $O(d,1)$ $O(d-1,1)$ de Sitter spacetime $dS^d$ deSitter gravity
linear algebraic group parabolic subgroup /Borel subgroup flag variety parabolic geometry
conformal group $O(d,t+1)$ conformal parabolic subgroup Möbius space $S^{d,t}$ conformal geometry conformal connection conformal gravity
supergeometry super Lie group $G$ subgroup (monomorphism ) $H \hookrightarrow G$ quotient (“coset space ”) $G/H$ super Klein geometry super Cartan geometry Cartan superconnection
examples super Poincaré group spin group super Minkowski spacetime $\mathbb{R}^{d-1,1\vert N}$ Lorentzian supergeometry supergeometry superconnection supergravity
super anti de Sitter group super anti de Sitter spacetime
higher differential geometry smooth 2-group $G$ 2-monomorphism $H \to G$ homotopy quotient $G//H$ Klein 2-geometry Cartan 2-geometry
cohesive ∞-group ∞-monomorphism (i.e. any homomorphism ) $H \to G$ homotopy quotient $G//H$ of ∞-action higher Klein geometry higher Cartan geometry higher Cartan connection
examples extended super Minkowski spacetime extended supergeometry higher supergravity : type II , heterotic , 11d

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

Urs Schreiber
(127.0.0.1)