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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

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Naber, "Topology, Geometry and Gauge Fields", especially the first volume, gives a very careful and detailed derivation of the properties of connections on principal bundles. It does not cover much ground but it is very thorough.

Madsen, Tornehave "From calculus to cohomology" covers quite a bit of material on vector bundles, connections and characteristic classes, but it is not an easy read.

The physicists oriented Nakahara, "Geometry, Topology and Physics" could also be useful.

Lawson and Michelsohn "Spin Geometry" is a very good reference, but it does not devote much space to theory of connections and bundles.

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