At the moment I'm trying to read Getzler/Berline "Heat kernels and Dirac operators", but I'm having a hard time to follow the rather brief outline of connections (of fibre bundles, principal bundles) etc. in the first chapter.
Can anyone recommend me a more detailed (modern) reference with careful explanation. (not Nomizu/Kobayashi).
I'm comfortable with connections, curvature etc in the context of Riemannian geometry, also with the topological theory of fibre bundles.
If you have a reference in mind, which does these things for general vector bundles, I'd be glad as well