As I understand it, Felix Klein sought to classify geometries with respect to what groups G that respected the structure of the given space X. Lately i read in an article on "the history of connections" by Freeman Kamielle that Cartan wished to generalize this notion. Is it correct to think of Fibre bundles ##(E, \pi, B, F)##= (whole space, projection, base, typical fibre) with a structure group G as the generalization that Cartan came up with? I.e. the whole space E does no longer respect the action of G, but each fibre respects it. Is that the correct understanding?