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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.
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