I am trying to understand about principal G bundle given a Lie group $G$. For that I started with action of Lie groups on manifold $M$ and convinced my self that if the action is smooth, proper, free then orbit space is a smooth manifold and corresponding projection map gives a fiber bundle $(M,\pi, M/G)$. We call this bundle and any other bundle isomorphi 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 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 satisfactory. Any suggestion regarding references are welcome.
Please do not add husemoller fiber bundles as it uses some other definition of 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 this 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.
Best Answer
"Foundations of Differential Geometry" by Kobayashi and Nomizu has a good introduction to principal bundles and connections on them. In fact, this is presented at the beginning and is used as the basis for their presentation of classical (in the present) differential geometry.