nLab torsion of a Cartan connection

for disambiguation see at torsion

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Definiton

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

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

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

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

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

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 $\mathfrak{g}/\mathfrak{h}$-valued parallel transport. In partciular then every point in the base manifold has a neighbourhood $U_x$ over which the given $G$-principal bundle trivializes and then this parallel trasport gives an identification $U_x \simeq U_{\sigma(x)}(G/H)$ with an open subset in the Klein geometry $G/H$.

Relation to integrability of $G$-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 $(H \to G)$ being the inclusion $O(d)\hookrightarrow Iso(\mathbb{R}^d)$ of the orthogonal group into the Euclidean group or the inclusion $O(d-1,1)\hookrightarrow Iso(\mathbb{R}^{d-1,1})$ of the Lorentz group into the Poincare group, then an $(H\to G)$-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.

Related concepts

• torsion of a G-structure

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)