nLab
torsion of a metric connection
Contents
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 X 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 ) ⋉ 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
Ω ∈ Ω 1 ( U , 𝔰𝔬 ( p , q ) ) \Omega \in \Omega^1(U,\mathfrak{so}(p,q)) (sometimes called the “spin connection ”);
E ∈ Ω 1 ( U , ℝ p + q ) E \in \Omega^1(U,\mathbb{R}^{p+q}) (sometimes called the “vielbein ”).
The metric itself is
g = ⟨ E ⊗ E ⟩ .
g = \langle E \otimes E \rangle
\,.
Accordingly also the curvature 2-form has two components:
R = d Ω + [ Ω ∧ Ω ] ∈ Ω 2 ( U , 𝔰𝔬 ( p , q ) ) R = d \Omega + [\Omega \wedge \Omega] \in \Omega^2(U, \mathfrak{so}(p,q)) – the Riemann curvature ;
τ = d E + [ Ω ∧ E ] \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 G subgroup (monomorphism ) H ↪ G H \hookrightarrow G quotient (“coset space ”) G / H G/H Klein geometry Cartan geometry Cartan connection
examples Euclidean group Iso ( d ) Iso(d) rotation group O ( d ) O(d) Cartesian space ℝ d \mathbb{R}^d Euclidean geometry Riemannian geometry affine connection Euclidean gravity
Poincaré group Iso ( d − 1 , 1 ) Iso(d-1,1) Lorentz group O ( d − 1 , 1 ) O(d-1,1) Minkowski spacetime ℝ d − 1 , 1 \mathbb{R}^{d-1,1} Lorentzian geometry pseudo-Riemannian geometry spin connection Einstein gravity
anti de Sitter group O ( d − 1 , 2 ) O(d-1,2) O ( d − 1 , 1 ) O(d-1,1) anti de Sitter spacetime AdS d AdS^d AdS gravity
de Sitter group O ( d , 1 ) O(d,1) O ( d − 1 , 1 ) O(d-1,1) de Sitter spacetime dS d dS^d deSitter gravity
linear algebraic group parabolic subgroup /Borel subgroup flag variety parabolic geometry
conformal group O ( d , t + 1 ) O(d,t+1) conformal parabolic subgroup Möbius space S d , t S^{d,t} conformal geometry conformal connection conformal gravity
supergeometry super Lie group G G subgroup (monomorphism ) H ↪ G H \hookrightarrow G quotient (“coset space ”) G / H G/H super Klein geometry super Cartan geometry Cartan superconnection
examples super Poincaré group spin group super Minkowski spacetime ℝ d − 1 , 1 | N \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 G 2-monomorphism H → G H \to G homotopy quotient G / / H G//H Klein 2-geometry Cartan 2-geometry
cohesive ∞-group ∞-monomorphism (i.e. any homomorphism ) H → G H \to G homotopy quotient G / / H 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
Last revised on December 18, 2014 at 14:44:39.
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