  As far back as the 1920s, the great French mathematician and geometer Élie Cartan had recognized that the notions of parallelism and affine connection admit of an important generalization in the sense that (1) the spaces for which the notion of infinitesimal parallel transport is defined need not be the tangent spaces that intrinsically arise from the differential structure of a Riemannian manifold \(M\) at each of its points; rather, the spaces are general spaces that are not intrinsically tied to the differential manifold structure of \(M\), and (2) relevant groups operate on these general spaces directly and not on the manifold \(M\), and therefore groups play a dominant and independent role. 