TITLE: Is there a book on differential geometry that doesn't mention the notion of charts?
QUESTION [8 upvotes]: 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..

REPLY [15 votes]: For Riemannian geometry (and therefore no gauge theory or Hamiltonian mechanics), I recall two beautiful coordinate-free expositions:
1)  Milnor's monograph "Morse Theory" has all of the essentials presented elegantly in one very short chapter.
2) Cheeger and Ebin's book, "Comparison Theorems in Riemannian Geometry" also does everything without coordinates.
For $U(1)$ gauge theory (i.e., Maxwell's equations), see
Maxwell's equations and differential forms
I would add that it was for me very difficult to learn well coordinate-free differential geometry without also grinding through a lot of messy calculations in coordinates. Learning coordinate-free differential geometry is like learning linear algebra using abstract vector spaces. As beautiful as that is, it is hard to appreciate without first learning linear algebra on $\mathbb{R}^n$.