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.

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

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



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