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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..

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    $\begingroup$ Just for the record, FWIW, I personally think the question is just fine here on MathOverflow $\endgroup$ – Yemon Choi Oct 20 '17 at 11:15
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    $\begingroup$ Basic differential geometry can be phrased in a coordinate-free way but when you need to prove/check something you will find that coordinate computations are often shorter and easier, hence most (if not all) textbooks use both. You may like Chapters 1-2 of "Einstein manifolds" by Arthur L. Besse. $\endgroup$ – Igor Belegradek Oct 20 '17 at 11:39
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    $\begingroup$ Dear @AlexM. My point had absolutely nothing to do with sheaves. It is more that I find myself needing "undergraduate level mathematics" (for certain values of undergraduate that really really really really are not as universal as some people seem to believe) in the course of actual research I am doing. It is also often the case that I don't immediately know the right words to be looking up or the right books to be reading, because if I did I wouldn't be asking online, I would be reading those books $\endgroup$ – Yemon Choi Oct 20 '17 at 14:36
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    $\begingroup$ I am very surprised to read what the OP wrote about Milnor's book. It is widely regarded as one of the best books of geometry ever written. Robert MacPherson described it as "unimprovable". I understand it is a matter of taste, but please try to be diplomatic. $\endgroup$ – Ben McKay Oct 20 '17 at 20:09
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    $\begingroup$ I'll chime in with Deane, to say I am stunned, and in fact disgusted, by this pronouncement on Milnor's wonderful book (one of my favorites even though it is well outside the fields where I do research). I would strongly advise removing it. $\endgroup$ – Todd Trimble Oct 20 '17 at 23:16
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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 $\mathbb{R}^n$.

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    $\begingroup$ I would add that although these books are co-ordinate-free in the sense that (mostly) the statements of main lemmas and propositions do not mention charts, the proofs of the lemmas and propositions often do involve charts. Indeed, this is hard to avoid: for example, how can you prove the existence of integral curves or geodesics without taking a chart and applying a theorem on the existence of solutions to systems of ODEs? $\endgroup$ – macbeth Oct 20 '17 at 17:12
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    $\begingroup$ I would also add that almost any modern book on Riemannian geometry is co-ordinate-free in the same sense as these two books. These two are the classics, perhaps, but there are more beginner-friendly books, Lee's for example, that take the same approach more slowly (because they're not racing to get to more advanced material). $\endgroup$ – macbeth Oct 20 '17 at 17:14
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    $\begingroup$ Lang's book, Linear Algebra, seems to do it abstractly. Almost any abstract algebra book will also do it that way. Books on functional analysis, too. $\endgroup$ – Deane Yang Oct 20 '17 at 17:27
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    $\begingroup$ I have to agree with macbeth. I haven't looked closely at more modern books, but it's hard to imagine that they don't discuss the coordinate-free approach in detail. Besides Lee, I suggest looking at the book by Gallot, Hulin, and Lafontaine. $\endgroup$ – Deane Yang Oct 20 '17 at 17:29
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    $\begingroup$ Both books deduce properties of the geodesic equation and the Jacobi equations from their description in local charts. $\endgroup$ – Igor Belegradek Oct 20 '17 at 20:23

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