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 $\mathfrak{iso}(p,q)$.

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

• integrability of G-structures

Related concepts

• first-order formulation of gravity

geometric context	gauge 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

References

Monographs:

• Richard W. Sharpe, Differential geometry – Cartan’s generalization of Klein’s Erlagen program, Graduare Texts in Mathematics 166, Springer (1997) [ISBN:9780387947327]

• Andreas Čap, Jan Slovák, chapter 1 of: Parabolic Geometries I – Background and General Theory, AMS (2009) [ISBN:978-1-4704-1381-1]

Discussion of torsion in gravitational classical field theory:

• Friedrich W. Hehl, Yuri N. Obukhov, Élie Cartan’s torsion in geometry and in field theory, an essay, Annales de la Fondation Louis de Broglie, 32 2-3 (2007) 157-194 [pdf, arXiv:0711.1535]

Discussion with an eye towards torsion constraints in supergravity:

• John Lott, The Geometry of Supergravity Torsion Constraints [arXiv:0108125]

  following:

  John Lott, Torsion constraints in supergeometry, Comm. Math. Phys. 133 (1990) 563-615 [doi:10.1007/BF02097010]