I am trying to understand about principal G bundle given a Lie group $G$. For that, I started with the action of Lie groups on manifold $M$ and convinced myself that if the action is smooth, proper, and free then orbit space is a smooth manifold and the corresponding projection map gives a fiber bundle $(M,\pi, M/G)$. We call this bundle and any other bundle isomorphic to this bundle a principal $G$ bundle. Only to understand just this it took so much time for me as there are many books with different definitions of principal bundles.
I am comfortable with this definition of a principal bundle and would like to know more enough to start reading connections on principal bundles. So, I would like to ask for your suggestion regarding some reading materials online on principal bundles. I checked it but did not find it satisfactory. Any suggestions regarding references are welcome.
Please do not add husemoller fiber bundles as it uses some other definition of a principal bundle and not Kobayashi also. I am benifited very much with chapter on Lie group actions on manifolds in Lee’s Book Introduction to manifold. I wish he had published some notes on principal bundles as well. Please do let me know any references along these lines.
Edit: I am now able to understand roughly the concepts of connections, holonomy groups from Kobayashi. Any reference which can supplement this book is most welcome. Now I am reading curvature form and structure equation and do not really understand what they are saying.