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Is Fibre Bundles Cartan's Generalization of Klein's Erlagen Program?

  1. Mar 28, 2014 #1
    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?
  2. jcsd
  3. Mar 31, 2014 #2
    Not exactly. I tried to read a little bit about this sort of thing a while back, but I'm no expert on it, either. But it's definitely more than just a fiber bundle. There were lots of early examples of fiber bundles, like normal bundles (where, in my opinion, the concept of a bundle finds its best motivation because you are naturally lead there to study the twisting of a neighborhood of the embedding, which you can identify with the normal bundle), and Cartan's moving frames (frame bundles--the canonical example of a principal bundle) that evolved into the bundle concept.

    That wasn't Cartan's idea. Fiber bundles are how it's formalized, but the idea was really to model spaces locally on Kleinian geometries. So, it's generalizing differential geometry on the one hand and Kleinian geometries on the other hand.
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