for disambiguation see at torsion
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
is the Lie algebra valued curvature form, then the torsion form is
e.g. (Sharpe 97, section 5.3, below def 3.1, Cap-Slovák 09, section 1.5.7, p. 85)
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 $G/H$.
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.
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.
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
Last revised on January 28, 2021 at 13:45:24. See the history of this page for a list of all contributions to it.