What are some books/texts that use chart free coordinate free language for things otherwise written in a coordinate based formulation? I would like to learn about covariant differentiation, curvature, bundles, characteristic forms etc., but without any charts or local coordinates. The most important areas would be those with applications in physics like gauge theory or Hamiltonian mechanics, where I so often hear all mathematical literature uses coordinate free language, but I never seem to find any such text. They say that they are doing calculations using intrinsic methods; it makes you wonder where all the tedious coordinate manipulations went.

Milnor's monograph "Morse Theory for example is a horrible book written in a really bad prosaic style , baez's gauge fields knots and gravity and Mallios's modern differential geometry in gauge theories are the kind of material im interested in. Baez is awesome up untill the point he decides something is too abstract and breaks it down in a chosen basis..

I wouldn't be asking online, I would be reading those books $\endgroup$ – Yemon Choi Oct 20 '17 at 14:36