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Good books of fiber bundles

  1. Jan 13, 2010 #1
    Becuase of geometric phase,I'm looking for a good book on fiber bundles, with a minimum of prerequistes and that takes a more intutive rather than formal approach.I am reading a book called modern differential geometry for physicists. It is a good book but sometimes abstract.

    I know about calculus of variations, multivariable calculus, vector calculus, linear algebra,and a bit of differential geometry.
     
  2. jcsd
  3. Jan 13, 2010 #2
    thsis is a broad subject. What in particular do you want ot learn about?
     
  4. Jan 13, 2010 #3
    thanks a lot. I want to learn about principle bundles, associate bundles ,vector bundles , parallel transport and homonomy in them.
     
  5. Jan 13, 2010 #4

    Fredrik

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    I recently finished Chris Isham's book "Modern differential geometry for physicists", and it covers the topics you're interested in (after going through the basic stuff about manifolds and Lie groups). All things considered I'd give it 4 stars out of 5, but I should warn you that the explanations are getting much more difficult to follow towards the end, where he talks about connections on principal bundles. It was almost like he had been in a hurry to finish the book. But everything before that is very good, and I was able to get through the difficult parts too, even though it was a bit painful.

    I should probably also warn you that there isn't much about holonomies in the book (since you mentioned that specifically), so if that's something you want to study in depth, you will have to do it somewhere else (but this book can at least prepare you for that).

    Another thing I found a bit annoying is that even though his goal is to teach you the stuff you need to know to understand that a Yang-Mills potential is a local representative of a connection on a principal bundle, and that kind of stuff, he never actually writes down the Yang-Mills Lagrangian and connects it to the stuff you've learned.
     
  6. Jan 14, 2010 #5
    Thank you very much!You are so kind and helpful.I am reading it now. Maybe it's the best book about this subject. elegant and graceful.I will try my best to finish it!
     
  7. Jan 14, 2010 #6
    I do not know an easy book. Perhaps Fredrik's book is the one you should use.

    Singer and Thorpe's Lecture Notes on Elementary topology and Geometry discusses geometry on oriented surfaces from the point of view of connections on circle bundles ( a circle bundle is a principal bundle). It is incredibly well written. The case of two dimensional Riemannian geometry is intuitive and has helped me to understand higher dimensional geometry. This book also introduces De Rham theory on manifolds, and covering spaces. Singer and Thorpe are both well known mathematicians (Singer is the Singer of the Atiyah Singer Index Theorem for Elliptic Operators. Thorpe is a well known Geometer) and wrote this book to introduce geometry from the modern view point. I can not recommend this book highly enough. Many of the mathematical tricks used in proofs are also modern and will serve you faithfully in higher dimensions.
     
  8. Jan 14, 2010 #7
    Thanks a lot!I read it just now.I think it may be a good reference to me!But in my believe I lack the ability to read it from the beginning to the end,because it was a litter hard for me.
     
  9. Jan 15, 2010 #8

    Fredrik

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    The book "Gauge fields, knots and gravity" by Baez & Muniain looks very interesting too. I haven't read it, but I hope I'll have time for it later this year.
     
  10. Jan 17, 2010 #9
    Thanks a lot! It is really a good book. And i think it is hard for me to read.
     
  11. Jan 17, 2010 #10

    DarMM

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    It's basically the best follow on from Isham's book. It deals specifically with the Yang-Mills stuff that you want.
     
  12. Jan 18, 2010 #11
    Thanks u very much, I think it intrigue my interest to read it.I want to study about quantum field. I think your suggested book maybe a good guide for me to study further.
     
  13. Jan 18, 2010 #12
    This book is basic and meant for undergraduates. You should try it.
     
  14. Jan 19, 2010 #13
    Thanks all the same! I think it is a fundamental book.I will study it later when I have time. Now I must focus myselft on the use of fiber bundle because my major is physics.I leave the complex and precise work to mathematician.
     
  15. Jan 19, 2010 #14
    mistake
     
  16. Jan 21, 2010 #15
    I borrowed the book today from library. It is really a good book!easy to read and insightful enough for physicists. Thank you a lot! I think I will enjoy it. By the way it is a good book for beginners to learn differential geometry.
     
  17. Jan 21, 2010 #16
    I found the series 1 last time, so I committed a mistake. You urged book is series 4.I found it in library by acctident. I think it is a wonderful book. Thank you very much. I will read it carefully. Easy to read and insightful book!It is also appropriate for beginners.
     
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