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Book on differential geometry

  1. Feb 6, 2013 #1
    Hi there. I want to learn some differential geometry on my own, when I find some time. My intention is to learn the maths, so then I can get some insight, and go more deeply on the foundations of mechanics. I need to start on the basics. I had some notions on topology when I did my analysis II course, but there were only rudiments. So I would like some text that starts really on the basis, and that could serve when I go into a text on geometric mechanics.

    Any idea if there is some text book like that with worked examples and that kind of stuff?

    Thank you in advance.

    If this isn't the proper section for this kind of question, please move it.
  2. jcsd
  3. Feb 6, 2013 #2
    Start with "Fundamentals of Differential Geometry" by Serge Lang.and go to "A Comprehensive Introduction to Differential Geometry" by Michael Spivak.
  4. Feb 6, 2013 #3
    Thank you.
  5. Feb 6, 2013 #4
    Spivak is probably overkill for what you want (it's a 5 volume series). Since it sounds like you're more interested in applications, try "The Geometry of Physics" by T. Frankel and Arnol'd's mechanics book (I forget exactly what it's called). These books introduce differential geometry and the applications.
  6. Feb 6, 2013 #5


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    This is a joke right? Are you actually trying to help the OP learn or just throw really hard books for no reason at someone looking for an introduction?
    As for the OP, I would suggest, if you are ok with anachronisms, Do Carmo's Differential Geometry of Curves and Surfaces and a natural order for getting up to Riemannian Geometry could be John Lee's series: Topological Manifolds then Smooth Manifolds then Riemannian Manifolds if you want a more comprehensive understanding than what is provided in say Frankel or Nakahara.
  7. Feb 6, 2013 #6
    This is about the worst advice I have ever seen on this forum. No kidding, it really is the worst.
  8. Feb 7, 2013 #7
  9. Feb 7, 2013 #8
    I know, right? I can't believe someone would assume the OP has a background in math, just because they asked about an advanced mathematical subject! The nerve of some people...

    All sarcasm aside, I realize the OP said that they wanted to learn the subject because of it's applications to physics. I don't think that's a fair reason to presume anything about their background in pure math, though, especially since the first volume of Spivak's books is a standard introductory textbook for differential geometry.
  10. Feb 7, 2013 #9


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    Who was talking about Spivak? The person recommended Lang's book as an introduction to differential geometry. I don't know if you are being pretentious or not but I have not met anyone who would ever recommend Lang's book as a first exposure to differential geometry. Maybe if you actually read the OP's post you would see he/she said he/she knew only the rudiments of topology and wanted a text that started on the basis. Try not to be so presumptuous next time and actually read the OP's requests.
  11. Feb 7, 2013 #10
    Did you actually read Lang?? The OP mentioned specifally that he had "some notions of topology from an Analysis II course". Do you think that that is enough to read Lang??

    I get the impression that you never even looked at Lang's book. In that case, I don't think you have the right to criticize anything.
  12. Feb 7, 2013 #11


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    Two undergraduate texts that offer entrée to differential geometry are: 1) Singer & Thorpe; "Lecture Notes On Elementary Topology And Geometry" covering topology, an introduction to manifolds, some Riemannian geometry and algebraic topology. Pressley; "Elementary Differential Geometry" covers geometry of curves and surfaces in three-dimensional space using calculus techniques.
  13. Feb 7, 2013 #12
    I was referring to Spivak's book. Yes, I'll admit I glossed over the Lang suggestion. Apparently that's the one that has you guys so pissed off, and I admit I'm not familiar with that book at all. When OP said "Analysis II", I assumed he meant something covering the material in Spivak's own "Calculus on Manifolds", which is perfectly adequate preparation for his differential geometry book.

    Did it ever occur to you that maybe he meant to say start with Spivak and then go to Lang, and just got them mixed up? No idea if that's in fact the case, but it seems likely based on what you're saying about Lang's book.
  14. Feb 7, 2013 #13
    He specifically said to start with Lang and then go to Spivak.
  15. Feb 7, 2013 #14
    The fact that the poster is banned right now should give away his intentions...
  16. Feb 7, 2013 #15
    I know, but I was thinking he might have gotten them mixed up. That seems like a more likely explanation than that he was somehow trying to impress the Internet by suggesting a high level book as an introduction :wink:.

    Anyways, I'm sorry for being sarcastic in my original post, but I didn't think a poor book suggestion deserved such open hostility.
  17. Feb 7, 2013 #16
    You're right, a poor book suggestion doesn't deserve hostility. But the poster was trolling. I don't react kindly to trolls. In the highly unlikely situation that the poster was not trolling, I apologize.
  18. Feb 7, 2013 #17
    Fair enough. Still, what's so unlikely about them meaning to say "start with Spivak and then go to Lang" and getting mixed up? I'm not trying to be argumentative (no, really :tongue:), I'm genuinely curious what you think. Especially since they're apparently now banned...
  19. Feb 7, 2013 #18
    There's nothing unlikely about it. But he didn't say it, and neither did he correct himself after we commented (although he could have been banned by then).

    In either case, even after you read Spivak, I would still not recommend Lang. Lang is far too abstract to be of any use for a textbook. There are much better books out there than Lang. Certainly for somebody into physics.

    Also, I kind of feel that Spivak is also a bit difficult for a first introduction to diff geo. It's completely right that calculus on manifolds is a sufficient preparation and that mathematically, he can handle it. But still, I think that your first encounter with diff geo should be with some text such as Do Carmo, O Neil or Pressley. Immediately going to the abstract general manifolds is a bad idea.
  20. Feb 7, 2013 #19
    Alright, that makes sense. Like I said, I was just annoyed about the level of hostility towards him - it didn't really seem appropriate, and it's not the first time I've seen something like that on this forum. Sorry about the sidebar, I'll let you all get back to your suggestions. :smile:
  21. Feb 8, 2013 #20
    Thank you all for your recommendations.
  22. Mar 1, 2013 #21
    I'm reviving this thread because I am in the similar predicament as the OP: physicist that wants to self-learn the basics without spending too much time decoding lots of rigor over vacations, for its mechanics and GR applications. I've already taken a basic GR course for math undergrads but it didn't cover much, so I'd like to learn all the aspects relevant to physics (Hamiltonian mechanics, dynamical systems etc) properly.

    I'm almost decided on getting Pressley's book because I've thumbed through it, liked the modern exposition, and can get it for really cheap. I think I could go through it pretty quickly as it seems accessible, but I've also checked out the one by Kühnel (seemed ok but has less problems) and Kreyszig (notation looked tiny and tedious).

    The only thing that holds me back is that Pressley only deals with Euclidian 3D spaces, is this too limited for GR and dynamical applications or will it suffice? If someone has a better recommendation I'm willing to hear it, but please bear in mind I haven't taken topology.
  23. Mar 1, 2013 #22


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    Pressley won't help you for GR because it deals with classical geometry of curves and surfaces embedded in 3 - space. It doesn't go into the details of smooth manifolds. Unlike Kreyszig's amazing functional analysis book, his diff geo book is terrible. Don't use it man just run away from it. You not having taken topology is only an issue when it comes to rigorous textbooks on the subject of differential topology (e.g. Lee - Smooth Manifolds) but if you just want to know enough to learn GR then why not just look at an advanced GR book? Wald, for example, develops everything you need to know in the text + appendices. This is pretty much as close as you can get to an exposition of differential geometry in a general manifold setting without going to a rigorous book on smooth manifolds (to my knowledge the only good, relatively popular book that is even less relaxed than Lee's book on smooth manifolds is the one by Loring Tu).

    EDIT: I just saw your other request regarding dynamics. From my experience, the books that develop classical mechanics in the manifold setting in a proper, rigorous way actually require more manifold theory than introductory GR (e.g. first 6 chapters of Wald). The book by Singer "Symmetry in Mechanics" tries to introduce symmetries in the manifold setting but in a way that doesn't assume much math (she doesn't properly define topological manifolds and is pretty hand wavy on lie groups and lie algebras and group actions and their orbits even though she then uses these concepts to talk about conserved systems) however, as noted, the lack of mathematical rigor will leave you unsatisfied and/or confused in my experience.
    Last edited: Mar 1, 2013
  24. Mar 1, 2013 #23


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    I liked Crampin and Pirani , as well as Fecko. Both are very physicky. Fecko is a little long winded, but I find all his asides charming and insightful.

    Lee is very, very good. Even though it's apparently a serious maths book, it's written so that even non-mathematicians can enjoy it at our own lower level (like a novel).
  25. Mar 1, 2013 #24
    Yeah I figured Carroll (which I've used) and Wald/MTW probably have everything I'll ever need for GR, but I also wanted to get a slightly more general understanding of the math subject, not just for GR. Namely the concepts in analytical and stat mech like phase plots, Liouville's & Noether's theorem, etc. Apparently to really understand these things fully one needs DG.

    I've already got a good calc of variations book and a book on dynamical systems on my reading list, but I wanted to read something short and a little more math pure beforehand.
  26. Mar 1, 2013 #25


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    Yes, mechanics at that level does require a really good understanding of differential topology as I noted above in the edit. If you want to then look at classical field theory in the diff geo setting it gets crazier IMO (well it was crazy for me anyways =p). This is a standard classical field theory text that uses diff geo: https://www.amazon.com/Classical-Covariant-Cambridge-Monographs-Mathematical/dp/0521675774 if you want a perspective on things although it is probably more extensive than other texts out there.

    In my honest opinion, pure math is best learned from math texts. As atyy noted, Lee is a book even physicists can enjoy (although it does assume some pre requisite topology knowledge, basically up to things like connectedness, path connectedness, compactness, para - compactness, properties of topological manifolds etc. to get down the core material in the text and then some of the more advanced chapters will require some knowledge of homotopy and, more prominently, knowledge of homology esp. if you want to learn about de Rham cohomology; there may be some other advanced things the later chapters require - I haven't ventured too far into the later chapters but you won't have to read the entire text if your goal is to eventually get to the physics). Go at it mate =D.
    Last edited by a moderator: May 6, 2017
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