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.
- How do principal bundles appear "naturally"?
- 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?
- 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?
- 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.