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Ehresmann connections; covariant derivatives; connections on vector bundles, principal bundles, ∞-bundles, submersions, bundle gerbes; holonomy and higher holonomy; parallel transport; torsion; curvature. See also the tags [principal-bundles], [vector-bundles], [gerbes], [curvature], [geodesics], [characteristic-classes], [torsion].

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Torsion-free Cartan connections

There are great answers about the torsion of linear connections and G-structures in What is torsion in differential geometry intuitively? …
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