References for Differential Geometry

Question

I would like to learn the language of Differential Geometry formally. I have a basic understanding of Non Abelian Gauge Theory from my study of physics. However I don't understand the language used by mathematicians(and physicists) properly. I am looking for books that start with basic notions In differential geometry, principle g bundles, connection, curvatures etc. After that I would like to learn about about sheaves, Cech cohomology etc. 

I would like the presentation to be sufficiently abstract while maintaining its root in structures used in physics. I am not interested(At this moment) in the most abstract way to treat the subject. 

I would eventually like have the knowledge to understand articles on N Lab(and references linked) and lectures such as Lectures on Higher structures In M theory to name a few. 

Feel free to recommend books that would be useful for students of different levels I suspect that people with varying backgrounds would be interested in studying these topics. It would be nice to have a compilation of books that teach this aspect of mathematical physics. 

Comments

The book by Marsden an Ratiu (together with its internet supplement) teaches the standard differential geometric language and is strongly physics-oriented. It doesn't go far enough, though, for your ultimate goals (no sheaves etc.).

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@ArnoldNeumaier Thanks for the reply. I am looking for a different kind of book at this point in my study. The book suggested looks good but it is motivated by examples in Classical mechanics,(In the spirit of V I Arnold's book) and it does not mention gauge theories. I do have a understanding of basic differential Geometry(Vector fields, forms, integration stokes theorem etc). While there are topics in this book that I would definitely like to study at some point. At this moment I am looking for a book specifically to study Gauge theories and Its connection to topology.(Monopoles, instantons etc..). 

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Try Naber Topology, Geometry and Gauge fields a two volume set. Baez Gauge fields knots and Gravity also contains some of this stuff

Answer 1

General differential geometry:

Boothby and Lee's 3 books (the last one is about algebraic topology). For going a bit deeper into algebraic topology and get some glimpses of sheaves I also recommend Shastri's book. Then you can read any classic text in algebraic geometry (assuming you know basics of commutative algebra) like Hartshorne (especially the first 2 chapters). These should give you more than enough background. One interesting book I am hopping to dig into lat some point is Kashiwara and Schapira  called Sheaves on Manifolds.

Answer 2

Here are some books which talk about some of the topics mentioned

1. Topology and Geometry for Physicists - Siddhartha Sen and Charles Nash
2. Geometry, Topology and Physics - Mikio Nakahara
3. The Geometry of Physics: An Introduction -  Theodore Frankel 

Comments

I am currently reading "The Geometry of Physics", and I like its style of giving many examples for the concepts introduced, visualizing things by clear figures, and giving proofs of the theorems mentioned (or if the proof is too extended it gives references). For me personally, it is also nice to read, as it is not written in a very terse definition-theorem-proof fashion but contains enough explanatory text. Some people might consider it to be too "wordy" though (?).

Concerning the topicality, Frankl definitively contains not exclusively differential geometry in a narrower sense, but features also some group theory, etc. However, as he explicitely says in the introduction, things like Kähler manifolds, the algebraic geometry used for string theory, or infinite dimensional manifolds are not considered.

So this books is probably not sufficient to understand the lecture notes mentioned.

Answer 3

I keep developing lecture notes that develop differential geometry from first principles, but such as to naturally grow into the discussion of higher structures in string theory and M-theory.

There is an online version of these lecture notes here:

Geometry of Physics

and there is a pdf version, this constitutes section 1.2 in the book

Differential cohomology in a cohesive topos

These notes are still being developed, but the first sections are stable and are really elementary.

By the way, I just gave a survey talk on

Higher Structures in Geometry and Physics

to a complete lay audience. Maybe you get something out of that.

Answer 4

A classic: Foundation of Mechanics, by Ralph Abraham and Jerrold Marsden. Free version here:

http://authors.library.caltech.edu/25029/