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Recommendations for diff geom. textbook

  1. Feb 24, 2007 #1
    I've been reading Nakahara as well as the sections of 'Superstring theory' by Green etc. on differential geometry in relation to all the methods used in string theory, particularly stuff on complex manifolds. However, while both of them are excellent introductions, they are somewhat devoid of exercises to actually test understanding.

    Does anyone have any suggestions on books which gives a decent number of exercises to work through on stuff like complex manifolds, cohomologies and holonomy?

    The Wiki page on holonomy uses this as a reference but I suspect it's not really pitched for people wanting exercises to check their understanding.

    Thanks :smile:
  2. jcsd
  3. Feb 24, 2007 #2
    Schutz's text Geometrical Methods of Mathematical Physics is a very nice text. That's what I'm using right now to solidify my background in the subject. However it is very light (or even devoid of) on complex manifolds, cohomologies and holonomy.

  4. Mar 3, 2007 #3
    There is also a nice Dover text called Differential Geometry by Erwin Kreyszig.

  5. Mar 5, 2007 #4
    I haven't read that book but I feel confident in saying that anybody who's serious about learning modern differential geometry should avoid Dover editions like the plague.
  6. Mar 5, 2007 #5
    I don't know why I didn't think of this before, but if you're looking for problems regarding complex manifolds, Steven Krantz's text Function theory of several complex variables (Wiley, 1981) has many good, specific problems that you can jump right in on. Krantz, although sometimes writing with a irritatingly pedantic voice, really knows how to write good problems. Wells' Differential Analysis on Complex Manifolds helped me get through grad school, but I can't remember if it had any problems in it. Structures on Manifolds by Yano and Kon has a couple of chapters on complex manifolds *with* a nice problem set.

    Re: cohomology. Shoehorn is generically right about Dover books. But there is a good one out there dealing with the relationship between topology and diffl. geometry: Goldberg's Curvature and Homology text.

    As for holonomy, I can't think of anything, except my own problem: Using the upper half-plane model of the hyperbolic plane, describe a pencil and paper method of determining the holonomy of a given non-ideal triangle. Show that there is a relationship between the holonomy and the area of the triangle. Generalize to any closed a.e. smooth curve.
  7. Mar 5, 2007 #6
    Thanks for the suggestion guys :)
    Are you sure that's published by Wiley, because it's not on their website and the Amazon page gives it as the American Mathematical Society. I've got £100 free with Wiley so it'd be nice to get that but they don't seem to be the publisher (unless it changed recently?).
  8. Mar 5, 2007 #7
    it might be out of print. but, yes, my library's copy is from Wiley.

    i just remembered too that john lee has still kept up his assignments from his complex manifolds course:

    these are mostly standard boiler-plate questions, but you may find them interesting.
  9. Mar 5, 2007 #8
    I'm studying out of Krantz's book right now and I'd say there's alot of excercises but I'm not sure that alot of them are complex manifolds ones. I'd be curious to read an example of a problem that he can just jump right in on.

    Let me warn anyone that might do problems from Krantz's book:
    Some of the problems, while well intended make no sense as stated. I think there is always something that you can do to modify the problem and make it a good problem. But you have to be careful.
    My professor (apparently one of Krantz's best friends) game me this advice:
    Listen to what he means and not to what he says!!

    That's been great advice so far. Once you do that, then yes the problems are great.
    My overall opinion about Krantz's book it that its great and I recommend it to anyone. Again that come with the warning that you must be careful. If you read something and it doesn't make sense, then think about what it would take to have it make sense and do that. It usually works.

    It has no exercises, but yeah, great book so far.

    As far as I recall, only the first edition was published by Wiley. The second by Brooks/Cole and a reprint of the second by AMS. My professor (same I mentioned above) said that the second edition is far better (in terms of much less errors) than the first. And this is taking into account the warning I gave above relating to the second edition.
    Last edited: Mar 5, 2007
  10. Mar 6, 2007 #9


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    here's a problem. prove the comple projective plane is a comple manifold. i.e. it has an open cover by three open sets, each holomorphically isomorphic to C^2, and natural holomorphic isomorphisms on overlaps of these sets.

    recall CP^2 is the quotient space of C^3-{(0,0,0)} by the equivalence relation, that parallel vectors are equivalent. I.e. if z is a non zero complex number , then (a,b,c,) and (za,zb,zc) are equivalent.

    Then prove CP^2 is compact as a manifold. then prove there are no global holomorphic functions on it. if you know what cech cohomology is, compute H^0(O) for it.

    then consider another example: look at the anaologous complex manifold CP^1, and in the product C^2 x CP^1, consider the set of points

    {(a,b),[z,w]): aw-bz = 0}. prove this is also a connected complex 2 manifold, and that the projection on the first coordinate, i.e. onto C^2, is almost an isomorphism, except over (0,0) there is a whole copy of CP^1.

    on the other hand the projection onto CP^1, is a locaL PRODUCT STRUCTURE. i.e. this set is actually a complex line bundle over CP^1, under that latter projection.

    if you know what this means, compoute the chern class, or just try to see if the latter projection has a global holomorphic inverse, i.e. a section.

    my point is that thinking and talking about complex manifolds is more fun than talking about books about them.
  11. Mar 8, 2007 #10
    And why is that? Some Dover texts are used as testbooks at University graduate schools (i.e. Northeastern University Physics Department). Some of them are superior text books. Do you know where these books come from? These are texts which were printed and used as texts in their day. When they go out of print for lack of numbers Dover will often by the copyright. This book on Differential Geometry is a very excellant book. Please don't bad mouth a text by its cover or publisher. Please at least take a look at the text before you pass judgement on it.

    Don't forget that the first papers on relativity are in a Dover book called "The Theory of Relativity" which includes Einstein's work in SR and GR!

    One top notch book on tensors etc is called Tensors, Differential Forms, and Variational Principles, by David Lovelock and Hanno Rund. This book is quite often recommended in the most modern textbooks on diff. geo. Its a great book too by my own experience of reading part of it (A solid read of the first few chapters)

    Last edited: Mar 8, 2007
  12. Mar 9, 2007 #11


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    i havent yet taken a course in diff geom, but i gather that the 5 volumes by spivak constitute a complete introduction to diff geom.
  13. Mar 9, 2007 #12


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    yes but they take a long time to read.
  14. Mar 10, 2007 #13
    I'll take you up one this in about two weeks (once winter quarter ends).
  15. Mar 10, 2007 #14


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    do you know a good book which doesnt consume your time?!
  16. Mar 10, 2007 #15


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    “Ptolemy once asked Euclid whether there was any shorter way to a knowledge of geometry than by a study of the Elements; whereupon Euclid answered that there was no royal road to geometry.”

    Fits perfectly with the topic. :biggrin:
  17. Mar 10, 2007 #16


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    Tell me about it!

    Not only that subject, but all subjects are fun to talk about and explore. It's so hard not to like talk about it because none of your friends know anything about it. It kills me inside. I'm dying to go to graduate school, and finally be able to get into hours of great conversations.

    I get to talk to one of my professors a lot though. He lets me stop by and stuff. Not so much now because he's really busy so I rarely see him, but that'll change next month (2 professors left for research). He enjoys sharing things with me and stuff. It's priceless. I owe a lot to him.
  18. Mar 15, 2007 #17
    Why? Since you admit to not reading it and Dover is popular for for having very good science books (some of which have been used as textbooks in some grad school, e.g. Northeastern University). What Dover books have you read to make such a wide sweeping assumption? I've used these texts for close to 20 years and have nothing but good things to say about them.

  19. Mar 20, 2007 #18
    The three-volume treatise by Dubrovin, Novikov, Fomenko "Modern Geometry - Methods and Applications" is one of the best texts on the subject.
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