Do we need a principle G-bundle to define a Chern number or is it enough to have a vector bundle? (and in the case this needs to have complex fibers?)
That is, is it mandatory to consider the action on the fibers of a Lie group?
I'm asking this because the wikipedia pages on Chern numbers do not mention the group action, but in the book in which I've found the proof that Chern number do not depend on the connection, the framework are principle G- bundles.
Hope somebody can answer my question!