torsion of a metric connection


Riemannian geometry

Differential geometry

synthetic differential geometry


from point-set topology to differentiable manifolds

geometry of physics: coordinate systems, smooth spaces, manifolds, smooth homotopy types, supergeometry



smooth space


The magic algebraic facts




tangent cohesion

differential cohesion

graded differential cohesion

id id fermionic bosonic bosonic Rh rheonomic reduced infinitesimal infinitesimal & étale cohesive ʃ discrete discrete continuous * \array{ && id &\dashv& id \\ && \vee && \vee \\ &\stackrel{fermionic}{}& \rightrightarrows &\dashv& \rightsquigarrow & \stackrel{bosonic}{} \\ && \bot && \bot \\ &\stackrel{bosonic}{} & \rightsquigarrow &\dashv& Rh & \stackrel{rheonomic}{} \\ && \vee && \vee \\ &\stackrel{reduced}{} & \Re &\dashv& \Im & \stackrel{infinitesimal}{} \\ && \bot && \bot \\ &\stackrel{infinitesimal}{}& \Im &\dashv& \& & \stackrel{\text{étale}}{} \\ && \vee && \vee \\ &\stackrel{cohesive}{}& ʃ &\dashv& \flat & \stackrel{discrete}{} \\ && \bot && \bot \\ &\stackrel{discrete}{}& \flat &\dashv& \sharp & \stackrel{continuous}{} \\ && \vee && \vee \\ && \emptyset &\dashv& \ast }


Lie theory, ∞-Lie theory

differential equations, variational calculus

Chern-Weil theory, ∞-Chern-Weil theory

Cartan geometry (super, higher)

For other notions of torsion see there.



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.


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 groupstabilizer subgrouplocal model spacelocal geometryglobal geometrydifferential cohomologyfirst order formulation of gravity
differential geometryLie group/algebraic group GGsubgroup (monomorphism) HGH \hookrightarrow Gquotient (“coset space”) G/HG/HKlein geometryCartan geometryCartan connection
examplesEuclidean group Iso(d)Iso(d)rotation group O(d)O(d)Cartesian space d\mathbb{R}^dEuclidean geometryRiemannian geometryaffine connectionEuclidean gravity
Poincaré group Iso(d1,1)Iso(d-1,1)Lorentz group O(d1,1)O(d-1,1)Minkowski spacetime d1,1\mathbb{R}^{d-1,1}Lorentzian geometrypseudo-Riemannian geometryspin connectionEinstein gravity
anti de Sitter group O(d1,2)O(d-1,2)O(d1,1)O(d-1,1)anti de Sitter spacetime AdS dAdS^dAdS gravity
de Sitter group O(d,1)O(d,1)O(d1,1)O(d-1,1)de Sitter spacetime dS ddS^ddeSitter gravity
linear algebraic groupparabolic subgroup/Borel subgroupflag varietyparabolic geometry
conformal group O(d,t+1)O(d,t+1)conformal parabolic subgroupMöbius space S d,tS^{d,t}conformal geometryconformal connectionconformal gravity
supergeometrysuper Lie group GGsubgroup (monomorphism) HGH \hookrightarrow Gquotient (“coset space”) G/HG/Hsuper Klein geometrysuper Cartan geometryCartan superconnection
examplessuper Poincaré groupspin groupsuper Minkowski spacetime d1,1|N\mathbb{R}^{d-1,1\vert N}Lorentzian supergeometrysupergeometrysuperconnectionsupergravity
super anti de Sitter groupsuper anti de Sitter spacetime
higher differential geometrysmooth 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 (