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Good Books to learn Holonomy

  1. May 26, 2012 #1
    I am trying to learn about Holonomy groups. My objective is to read Submanifolds and Holonomy from Berndt, Olmos and Console. I know I need a lot of prerequisites before understanding that book. So I started first reading Lee's book of Smooth Manifolds (I readed it until the Submanifolds Chapter and then I jumped to Tensor Theory). After that I started reading Lee's book of Riemannian Geometry (I have nearly completed it, I am in the Submanifolds Chapter).

    Until now, I have found all the concepts I learned from Lee's Riemannian Book very algebraic. I mean, I still don't understand the geometric value of the riemannian curvature endomorphism, the curvature tensor or the scalar curvature.

    I have tried now to pick a book about holonomy. I took from the library Joyce book:Riemannian holonomy groups and calibrated geometry. It had different definitions about connections and was very difficult to follow. I tried on Riemannian Geometry and Holonomy Groups, also difficult and different definitions.

    So my question is what do you suggest guys on reading now to try to understand holonomy? Maybe I need to read a more advanced book on riemannian geometry than Lee's book. I was thinking maybe on Spivak vol 1 & vol 2 or Foundations of Differential Geometry of Kobayashi and Nomizu. Or is there any book that explains about holonomy with the concepts I have learned until now?

    Thanks in advanced,

  2. jcsd
  3. May 27, 2012 #2
    why do you want to understand string theory without any basis to understand it? a rhetorical question, but not one which isn't seeking some answer. i can suggest any number of books that you can read, and in five years you can still be stuck in the purgatory land of Lie Groups, like i am. but i did find that the "introduction to differentiable manifolds" (MacKenzie, Dover) was very helpful to finally explain how to deal with the fibre bundle concept. from here, stepping past the Lie Groups again, one can see sight of land, parallel transport. to climb the mountain of holonomy i haven't even begun.
  4. May 27, 2012 #3
    No, my goal is not to understand string theory. In fact, I didn't know that you could apply it to string theory. For my master program in mathematics I must explain how the proof of Bergers Theorem using Extrinsic Holonomy theory was done in the article written by Carlos Olmos.

    I still have some months to do my work, but well as you say the mountain is very steep. Trying to understand parallel transport is giving me problems. So I intend another book now that I am finishing Lee's one. Will look the book you have suggested me tomorrow.

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