# Background to learn about manifolds

Greetings

As a necessity to go forward with physics, I find myself in the need to learn about manifolds. Being an engineering student, I don't have the chance to study topology in all its glory. So, can any one point me to the right direction, i mean things I must know first and the materials to fill the gaps before approaching manifolds?

Thanks

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I'd like to suggest you ask this question on math.stackexchange – Ryan Budney Apr 10 '11 at 19:24
Sorry, Didn't know they (mathsoverflow & math.stackexhange) are completely different. Thanks – Toussaint Apr 11 '11 at 1:06

That depends on your mathematical background and the level of abstraction you want.

The minimum amount of concepts you should be familiar with, ranges from topological spaces and its basic properties (elements of point-set topology) to multivariable calculus (covering implicit and inverse function theorems), passing by elements of multilinear algebra to manage tensors. A good mathematical physics book where you can find this and even more, like an introduction to differential geometry through manifolds, is Szekeres' "A Course in Modern Mathematical Physics".

If you need this little background in topology, very good and short books on general topology are Runde's "A Taste of Topology" for a more formal approach and Jänich's "Topology" for a more didactic and graphical-reasoned introduction. Then the approach to manifolds from pure differential topology could be started by Jänich - "Introduction to Differential Topology" which is very graphical, short enough and quick to the point to master fully.

But if you are really interested in these matters from a theoretical physics perspective, you should by all means read the book by Nakahara - "Geometry, Topology and Physics"; there you can read a very thorough introduction (although sometimes concise) to most of the topics of differential geometry and topology of interests for physics: homology and homotopy groups, calculus on manifolds, Riemannian geometry, complex geometry, fiber bundles, connections, characteristic classes, index theorems with applications. It is a wonderful book if you have enough background and use supplementary readings (for example in tensor, multilinear, algebra). Other similar book with the same spirit is Frankel's "The Geometry of Physics" which is longer dealing with most of the same contents more deeply; nevertheless I do not find it as useful and straight to the point as Nakahara's (and Frankel's notation seems to me not the most orthodox standard compared to other books I have used on the subject). Similarly, I recommend the new book by Eschrig - "Topology and Geometry for Physics" which again deals with the same content as Nakahara's but with a less mathematical exposition (definition, theorem, proof...) since it is written to be read as lectures or a physics text. However, it is very detailed and instructive anyway, uses many figures and develops the same amount of detail or more at some points. This kind of books develop the necessary topological background all along as needed so if you have enough background on vector analysis/multivariable calculus you can approach these books directly to learn about manifolds.

For a more mathematical purely formal treatment of differential geometry on manifolds I would dive in the wonderful book by Nicolaescu - "Lectures on the Geometry of Manifolds" since it is very complete and modern. Another slower mathematical exposition is Jeffrey Lee's "Manifolds and Differential Geometry" which may be useful to you as a companion to the other physics-oriented texts, since it develops many details and background at some points. They are both wonderful books in my opinion.

For an approach to manifolds through mechanics, the classic book of V.I. Arnol'd - "Mathematical Methods of Classical Mechanics" is great but it is very applied and focused on symplectic geometry (and despite being a masterpiece, it is a little bit out-dated in style for my tastes). Another differential-geometric introduction to mechanics is José/Saletan's "Classical Dynamics: A contemporary approach". The most advanced book of this kind is the bible by Marsden/Abraham - "Foundations of Mechanics" which I think is a masterpiece trying to develop the complete theory of classical mechanics (hamiltonian/lagrangian/Hamilton-Jacobi) from a purely differential-geometric perspective using manifolds.

My personal advise would be to read Eschrig alongside Nakahara (or Frankel depending on your likings of style), that is what I did with a background on theoretical physics. Then you should consult some of the other books to fill particular gaps and needed background or to use as companions for more examples or exercises.

This is a wonderful and very interesting subject, good luck!

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What an impressively rich and interesting list: thanks, Javier! – Georges Elencwajg Apr 11 '11 at 12:15
You are welcome! I may not know yet all the mathematics I want, but I do know the bibliography pretty well (as until now I have been learning by myself a lot of topics buying and reading books). – Javier Álvarez Apr 11 '11 at 15:57
That explains this beautifully detailed list!!! Thanks – Toussaint Apr 12 '11 at 9:38
I'm impressed,Javier-that's as good-if not better-a short bibliography of advanced mathematics texts as I've ever seen. It's also rather interesting how the old classics of Spivak, Guillimen and Pollack and Milnor are being seen less and less in such lists-and J.Lee's and Nicolaescu 's books are being seen more and more. It IS hard to find a source that reaches just the right balance of rigor,breadth and clarity-where a student can master a substantial portion of modern differential geometry in a reasonable amount of time.Those 2 are certainly excellent suggestions. – The Mathemagician Aug 13 '11 at 3:04

Dear Toussaint, Frankel has written a book whose very title The geometry of physics: an introduction makes me think it might be what you want. The second paragraph of the introduction confirms this impression:This is a textbook that develops some of the geometrical concepts and tools that are helpful in understanding classical and modem physics and engineering. (the last word will warm your heart!)

Tensors and exterior products, for example, are presented in a very concrete way, without introducing more abstract concepts like quotients of free modules. And the pictures are numerous and very evocative.
Google will let you browse the book

To mathematicians Like many of us, I have a melancholy love for physics from my student days and I find this book written in exactly the right language for a mathematician wanting to (re)learn some physics (the title doesn't say it but there is quite a bit of physics in this book)

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Javier doesn't seem to like the book as much as I do and this difference of point of view is of course quite healthy and welcome on this site. Since he mentions Frankel's notations, I would like to add that on my part I find them quite orthodox, clear and even elegant. – Georges Elencwajg Apr 10 '11 at 22:59
I'll make sure I have it too. Javier has given a very exhaustive and detailed list. since it's not easy to choose, it's better then I have this one as you have mentioned it. Furthermore I'm more on the physics side than the pure mathematics side. Thanks – Toussaint Apr 11 '11 at 1:21
@ Georges Elencwajg: sure, Frankel's book is great and that is why I included it in my recommendations since I also bought it and used it. Nevertheless when I started to learn mathematics on its own I was coming from a background in general relativity and the like, so at that time Frankel's seemed to me a little bit harder than others like Nakahara and did not cover topics I liked like complex manifolds, index theorems and applications to gauge theory. But Frankel's is a GREAT book now that I know enough of the subject! – Javier Álvarez Apr 11 '11 at 7:52

I have really enjoyed browsing through Novikov and Taimanov. It starts out with very modest prerequisites and covers quite a bit of material.

Once you have the basics down I would recommend the new book by Peter Michor. It is one of the few sources I have found that "stresses naturality and functoriality from the beginning and is as coordinate-free as possible."

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Michor's book is really a very very good one. THough being quite formal, most of the stuff is really useful (which is not the case ina randam diff geom book at all!) and the proof are understandable but still very efficient. I used that book a lot when it was still a lecture note on his website... – Stefan Waldmann Apr 11 '11 at 7:48

You might like SICM (Structure and Interpretation of Classical Mechanics), http://mitpress.mit.edu/sicm/. It's not really about manifolds (discusses them some) but maybe it's the kind of physics you're looking for. Full text is online and I always like it when they do that.

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Thanks!! I have this one and it is exactly among the reasons why I have to understand manifolds. In the book, there's a focus on Lagrangian and Hamiltonian Mechanics which at some point, you have to know manifolds. And that's where I find myself stuck. It looks like great book! – Toussaint Apr 11 '11 at 8:08