What is a good reference to study Fiber bundles whose fibers are smooth manifolds (but not necessarily vector spaces)? I wish to look into details of such manifolds, the structure of coordinate charts, bundle morphisms, pullbacks etc. Any suggestion is appreciated. Thanks.
2026-03-29 20:09:41.1774814981
Reference for Fiber bundles with manifolds as fibers.
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N.Steenrod, "Topology of Fibre Bundles", Princeton University Press. Specifically, Part I of the book. It is a bit dated, but is very carefully written and covers all the basics that you are asking about. I also like the fact that Steenrod's book has the obstruction theory and characteristic classes done via the obstruction theory.
In case if you are interested in Ehresmann connections on general smooth fiber bundles, they are covered in
I.Kollar, P.Michor, J.Slovak, "Natural operators in differential geometry", Springer-Verlag, 1993.