nLab
torsion of a Cartan connection

Contents

for disambiguation see at torsion


Context

\infty-Chern-Weil theory

Differential cohomology

Contents

Definiton

Given an (HiG)(H \stackrel{i}{\to} G)-_Cartan connection_ \nabla on a manifold XX, its torsion is the component of its curvature in the coimage of i *:𝔥𝔤i_\ast : \mathfrak{h}\to \mathfrak{g}.

So if UXU \to X is a cover over which the underlying GG-principal bundle tricializes (for instance the total space of the GG-principal bundle itself if one models the Cartan connection by an Ehresmann connection) and

ωΩ 1(U,𝔤) \omega \in \Omega^1(U,\mathfrak{g})

is the Lie algebra valued curvature form, then the torsion form is

τcoim(i) *ωΩ 1(U,𝔤/𝔥). \tau \coloneqq coim(i)_\ast \omega \in \Omega^1(U,\mathfrak{g}/\mathfrak{h}) \,.

e.g. (Sharpe 97, section 5.3, below def 3.1, Cap-Slovák 09, section 1.5.7, p. 85)

Properties

Integrability of local charts

If the torsion of a Cartan connection vanishes, then it has flat 𝔤/𝔥\mathfrak{g}/\mathfrak{h}-valued parallel transport. In partciular then every point in the base manifold has a neighbourhood U xU_x over which the given GG-principal bundle trivializes and then this parallel trasport gives an identification U xU σ(x)(G/H)U_x \simeq U_{\sigma(x)}(G/H) with an open subset in the Klein geometry G/HG/H.

Relation to integrability of GG-structure

If the Cartan connection is regarded as providing, in particular, a G-structure, then the condition that its torsion vanishes is the integrability of G-structures.

Examples

(Pseudo-)Riemannian geometry

For (HG)(H \to G) being the inclusion O(d)Iso( d)O(d)\hookrightarrow Iso(\mathbb{R}^d) of the orthogonal group into the Euclidean group or the inclusion O(d1,1)Iso( d1,1)O(d-1,1)\hookrightarrow Iso(\mathbb{R}^{d-1,1}) of the Lorentz group into the Poincare group, then an (HG)(H\to G)-Cartan connection encodes a (pseudo-)Riemannian geometry with metric connection. Its torsion then is the torsion of a metric connection. See also at first-order formulation of gravity.

References

Textbook accounts include

  • R. Sharpe, Differential Geometry – Cartan’s Generalization of Klein’s Erlagen program Springer (1997)

  • Andreas Cap, Jan Slovák, chapter 1 of Parabolic Geometries I – Background and General Theory, AMS 2009

Discussion with an eye towards torsion constraints in supergravity is in

  • John Lott, The Geometry of Supergravity Torsion Constraints Comm. Math. Phys. 133 (1990), 563–615, (exposition in arXiv:0108125)

Last revised on December 21, 2014 at 18:25:58. See the history of this page for a list of all contributions to it.