# Torsion-free Cartan connections

Let $$M$$ a differentiable manifold and $$H\subset G$$ a Lie group with a closed subgroup such that $$G/H$$ is connected. A $$H\subset G$$-Cartan connection on $$M$$ can be defined by

• A principal $$G$$-bundle on $$M$$, we call it $$Q$$
• A $$G$$-principal connection on $$Q$$, we call its connection $$1$$-form $$\omega$$ and its curvature $$2$$-form $$\Omega$$
• A reduction of the structure group of $$Q$$ to $$H$$, under the form of a subbundle $$P$$ which is $$H$$-principal and on which $$\omega$$ in non-degenerate

I am trying to get an idea for the torsion of a Cartan connection associated to a $$H\hookrightarrow G$$-geometry. There are great answers about the torsion of linear connections and G-structures in What is torsion in differential geometry intuitively? in particular Mathieu ANEL's answer gives an interpretation as the translation component of the "Cartan curvature" form.

I am looking for a "geometrical" interpretation of the vanishing of torsion. Vanishing of the total Cartan curvature is equivalent to the Cartan geometry being "locally flat", that is, locally isomorphic to the reference homogeneous space [1,2]. That said I see on the ncatlab that vanishing of the torsion should be enough of flatness to have local trivialisations (it seems too strong but I may be missing a subtlety).

I tried to see what vanishing torsion implied on parallel transport. As written by Robert Bryant if $$\mathfrak h$$ is not stable under $$\operatorname{Ad}(G)$$ then vanishing torsion does not mean that $$\Omega$$ has value in $$\mathfrak h$$ on all of the principal $$G$$-bundle $$Q$$. That said, $$Q$$ comes with a reduction to $$H$$ which defines a section $$\sigma$$ of the fibre bundle $$Q/H$$.

Vanishing torsion means that $$\Omega$$ takes value in $$\mathfrak h$$ on $$P$$. It follows that for any $$x$$ in $$M$$, if we identify the fibre $$Q_x/H$$ with $$G/H$$ using $$\sigma(x)$$ as origin ($$[eH]\in G/H$$; there is $$H$$ worth of ambiguity) the representation of the curvature on $$Q/H$$ (a vertical vector field-valued $$2$$-form) takes value in $$\mathfrak h$$ at the origin, which means that it vanishes on $$\sigma(x)$$.

Are there any more geometrical conclusions to be drawn? I first mistakenly hoped that restricted holonomy would have to preserve $$\sigma$$ but according to the Ambrose-Singer theorem the infinitesimal holonomy at $$p$$ is generated by the values of the curvature at all points that can be connected to $$p$$ by a parallel path so that vanishing torsion does not seem to be sufficient.

 : Sharpe, R.W, Differential Geometry. Cartan's Generalization of Klein's Erlangen Program, Graduate Texts in Mathematics

 : Čap, Andreas; Slovák, Jan, Parabolic geometries I. Background and general theory, Mathematical Surveys and Monographs