Why do people care about principal bundles?

I’ve started to learn a little about principal bundles (in the smooth category) and while I see how notions like connections and curvature are abstracted from the setting of vector bundles and brought into the principal bundles world, I feel highly unmotivated regarding to why this is done in the first place.

So,

  1. How do principal bundles appear “naturally”?
  2. Why did people started thinking about connections, curvature, parallel transport on principal bundles? Why would one want to put connections on them? Study their curvature? Even determine whether they are flat or not?
  3. Do we gain something by considering the frame bundle associated to a vector bundle and studying its connection and curvature and not working directly with the vector bundle itself?
  4. What kind of problems the machinery of principal bundles helps to solve? What notions does it clarifies?

Feel free to provide examples of results or ideas from any field, including physics. Explicit geometric examples are especially welcomed.

Answer

For 1, given any free action of a compact Lie group G on a manifold M (or a free proper action of a noncompact Lie group), the orbit space M/G naturally has the structure of a smooth manifold such that the projection π:MM/G is a smooth submersion. (Free means the only group element which fixes at least one point is the identity).

It turns out that π is actually a G-principal fiber bundle. So, if you care about group actions on manifolds, principal bundles arise naturally.

For 3, one of the main uses of the frame bundle I know of is the following: Suppose a compact Lie group G acts effectively on a Riemannian manifold M. (Effective means the only group element which fixes all points is the identity). Since the action is not free, the orbit space M/G isn’t a manifold in any kind of natural way (though it is still not so bad as a topological space!).

On the other hand, the action induces an action on the tangent bundle TM (which still might not be free), and induces and action on the frame bundle FM. This induced action on the frame bundle is free, so the quotient FM/G is a manifold, so all the tools of differential geometry can be used to study FM/G, which in turn can give information about M.

Attribution
Source : Link , Question Author : levap , Answer Author : Jason DeVito

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