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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
    Commented Oct 20, 2017 at 11:15
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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
    Commented Oct 20, 2017 at 14:36
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    $\begingroup$ @macbeth: a smooth manifold is by definition a (second-countable Hausdorff) topological space equipped with a collection of charts whose transition functions are smooth That's one definition. Others are possible. Some are equivalent, some inequivalent. Differentiable manifolds are a subset of PL manifolds, which are a subset of topological manifolds. PL manifolds are defined in a way that's qualitatively different from your definition. Cf. math.stackexchange.com/questions/53021/… and arxiv.org/abs/1405.0984 . $\endgroup$
    – user21349
    Commented Oct 20, 2017 at 19:18
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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
    Commented Oct 20, 2017 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
    Commented Oct 20, 2017 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
    Commented Oct 20, 2017 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
    Commented Oct 20, 2017 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
    Commented Oct 20, 2017 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
    Commented Oct 20, 2017 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
    Commented Oct 20, 2017 at 20:23

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