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Please recommend differential geometry books

  1. Feb 4, 2005 #1
    I want to learn the mathematical language of Ashtekar's formulation / Loop Quantum Gravity in 3 months.

    Which introductory differential geometry texts do you recommend?
    I prefer books that are more mathematical, such as those that start from definitions, theorems, proofs, etc but not too comprehensive.

    Please also recommend books on mathematics beside differential geometry that are essential.

    I will appreciate
  2. jcsd
  3. Feb 5, 2005 #2


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    I'm not sure what background you have, but Baez and Muniain's "Gauge Fields, Knots and Gravity" is designed precisely to teach differential geometry with the aim of eventually getting to loop quantum gravity (the Ashtekar variables are introduced in the last chapter). It's also a very well-written book.

    After that, I imagine Rovelli's LQG book would be a good place to go, although I haven't read it myself.
  4. Feb 5, 2005 #3


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    when I was in school there were two general courses in differential geometry. Math 140 was for undergraduates and it was very concrete, dealing with curvature of paths and surfaces in 3D space. Math 240 was for graduate students and was more abstract and thorough.

    At the same time there were courses in General Relativity, at adv. undergr. and then again at grad student level.

    If I remember right, the math 240 textbook was Bishop and Crittenden and Gen Rel text was MTW (misner thorne wheeler)

    and then there is the issue of gauge theory and connections and bundles, which logically comes after a general course like 240.

    So far I have not recommended anything. I am just thinking.

    Why dont you do this: download the 90-page book called "Preparation for Gauge Theory" which is selfcontained, has all the necessary definitions, clearly written, and just see if you can read it?

    And also have a look at MTW if you have not already. I assume you are near a library and dont have to buy every book you want to try out.

    In any case Preparation for Gauge Theory is free and is arxiv.org if I remember correctly. Bootleg MTW may also be online too, but your library should have it.

    Please tell us more about your situation. Are you near a college or university library? Have you already taken undergrad Diff Geom and Gen Rel. courses? Are you on your own, or are you, for example, already a Physics major or Physics grad student at some university?

    So far I cannot offer any advice. The textbooks I used are probably out of date and out of print. Go to the student bookstore and see what they use nowadays for the Grad level Diff Geom course (like whatever corresponds to 240).

    Hopefully one or more other people will respond to your question on this thread. You should get several person's advice.
  5. Feb 5, 2005 #4


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    Hey, I just saw Stingray's post. Scratch what I said and buy the Baez!

    Baez is a really good writer. I am not familiar with his and Muniain
    "Gauge Fields...." book. But Stingray description and Baez being the author convinces me

    [edit: I went and looked up the page for this book at the publisher
    this is the World Scientific bookstore
    It is $46 in paperback edition. The webpage gives the table of contents.
    It looks like a good book. maybe worth owning if you cant easily come by a library copy]
    Last edited: Feb 5, 2005
  6. Feb 7, 2005 #5

    My background:

    Physics: I have learnt Special/General Relativity and Quantum Mechanics, not yet learnt Quantum Field Theory.

    Most of Linear Algebra, and algebraic Topology.
    group theory but up to topological group.
    some differential geometry including elementary differentiable manifold theory, Riemannian Geometry, but not differential forms.

    I am a Physics Graduate, in a University.

    Rovelli's Book seems too hard for me.
    Baez and Muniain's "Gauge Fields, Knots and Gravity" looks cool.
    Is it as hard as Rovelli's?
    This book is popular, I can't borrow it out from libary.

    It's too hard for me.

    MTW (misner thorne wheeler) is very good but very thick, I will take it as a reference.
  7. Feb 7, 2005 #6


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    For differential Geometry for a physicist the bible on the subject is Nakahara. For a mathematician something like say Spivak's series is very readable.

    Basically what you want to learn is elementary point set topology, and then generalized connections, lie groups, differential forms and then bundle theory. Later you'll need cohomology. All this stuff is in the aforementioned books. You won't understand anything in LQG unless you have that background.
    Last edited: Feb 7, 2005
  8. Feb 7, 2005 #7


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    No, Baez's book is much simpler than Rovelli's. I think it would fit in well with your background. Another book that might be good is Frankel's "Geometry of Physics." I've only flipped through it, but my mathematician friends say it's very good.

    I'm not a fan of Nakahara. I remember it being very boring and little more than a collection of definitions. I haven't looked at it in a few years, though, so maybe I just didn't know enough to appreciate it at the time.

    I'm not sure I'd recommend MTW either. It's very wishy-washy if you like a rigorous presentation (which you say you do). You also said that you want to learn things quickly, and MTW isn't exactly concise. For overall GR math, most other (upper level) GR books are much better IMO. LQG requires quite a few things that you won't find in any classical GR books, though.
  9. Feb 7, 2005 #8


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    Nakahara is good to have on your shelf, but unless you've already had a pretty complete intro to the field, it would be impossible. It's very much cram it all in as much as possible, and be rigorous, but forget motivation. To provide the necessary prequel to Nakahara, I'd recommend Nash & Sen, Topology and Geometry for Physicists. Don't let your thirst for rigor override your needs for clarity. "Suffivient unto the day is the rigor thereof."
  10. Feb 7, 2005 #9


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    Nash and Sen's book is also very good, though it does lead you wanting more IMO. Nakahara immediately made sense to me when I was a grad student, probably b/c I was more mathematically oriented. Sometimes physicist books just throw stuff out there without any definition and that can confuse me a lot more than the semi rigorous treatment. Its also a very good exercise in physicist notation that appears in the literature.

    The point I guess is if you understand Nakahara's treatment, you are almost there (but not quite).

    Again check out Spivak's books. They start out really easy, suitable for an undergrad, though it is a huge four volume set. As a grad student you really need that anyway, especially if you are going into theory work.

    I've always been of the mind that differential geometry is an extraordinarily simple subject once you get passed the notation. You then see it for what it is, a very simple game with a tremendous amount of depth and room for logical generalizations and added structure. LQG is just that, one way to pick structure on the theory.
  11. Feb 7, 2005 #10
    I agree with Haelfix. Just go straight to the mathematicians (Spivak) treatment. Trying to chince on rigor only comes back to haunt you either in having one hand tied behind your back or substandard papers.

    On a funny note, I was at a GR/Diff. Geo. conference the other day. You can tell right away who's a mathematician and a physicist by their 1) presentation or 2) the questions they ask the presenter.
  12. Feb 8, 2005 #11
    Thank you. I think I have found the books I want.

  13. Nov 26, 2011 #12
    In the post link below, I try to pull out the minimal set of math and physics ideas, backed up by the actual history, that underlie the knowledge needed to navigate from junior level math/physics through graduate school and beyond, including the current methods in theoretical physics.

    [Included are texts/references on algebra, topology, geometory and topology for the theoretical physicist in an uploaded Word document]. (The book by Nakahara is particularly good for guage field theories)

    I list and review a core set of the best, clearest books and literature to this end, often including what you should get from each book/article. I probably would have saved about a decade, and lots of money had I had a "syllabus" like this.



    A. Alaniz
  14. Nov 27, 2011 #13


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    your listing is not approved yet.
  15. Dec 14, 2011 #14

    You are such a strange creature that you created your own reading list and post it here and there. I don't think it interesting and necessary to market your own syllabus. Sometimes your syllabus can only be useful to yourself, while no one except you can profit from such a huge reading task. Farewell, interesting man.
  16. Dec 14, 2011 #15
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