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The last of the four definitions of a Cartan connection on the Wikipedia page describes it as an Ehresmann $G$-connection on the associated bundle $E=P\times_H (G/H)$. It claims that the Cartan connection on $P$ can be reconstructed using the usual association procedure. However, this seems wrong to me because for that the $G$-action on $G/H$ needs to be effective, which is not always true. So it seems that this definition works only for effective Cartan geometries.

Is this correct? And is there a general definition of Cartan geometry in terms associated bundles?

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    $\begingroup$ I think you are right: the Wikipedia authors are assuming that $G$ acts effectively on $G/H$. The existence and the uniqueness of the $H$-bundle giving rise to the $G/H$-bundle seems to me to be unclear. $\endgroup$
    – Ben McKay
    Commented May 9 at 10:30
  • $\begingroup$ For me, a Cartan geometry is just a "natural and reasonable" deformation of a homogeneous space of Lie type ( that's, the quotient of a Lie group by a closed subgroup, whatever the algebraic structure of the Lie supergroup), the deformation being given by a Ehresmann connection on the supergroup, with vertical vector bundle, the right translations of the Lie algebra of the Lie subgroup. $\endgroup$
    – Eric Arnéo Vespira Kengne
    Commented Sep 16 at 19:14

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