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I am finding it hard to understand in what way the Mauer-Cartan form $\omega_G$ of a Lie group $G$ can be used to define a connection on a bundle $G \to G/H$ in the same way that parallel transport of a section can.

Take for example the case of real projective space, where $G=PGL_n$ and $H$ is the subgroup that preserves the point $(1,0,...,0)$. Then $\mathbb{R}P^n$ is diffeomorphic to $G/M$ and $G$ is a principal H-bundle over $G/M$. This much is clear. I also know that the columns of a matrix $h \in H$ can be identified with a basis for the associated vector bundle over $G/M$. For vector bundles I am used to a connection being defined via the one-form

$\omega^i = dz^i + \Gamma^i_{\mu j} dx^{\mu} z^j$

or the parallel transport equation

$\dot{z}^i + \Gamma^i_{\mu j} \dot{x}^{\mu} z^j =0 \ ,$

but am having real difficulty seeing how, in the example described above, this relates to the notion of a connection defined in terms of the Mauer-Cartan forms of $G$ or $H$ (in the same way that parallel transport defines a connection on the bundle of linear frames). Any help would be much appreciated.

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