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The Euler class of the unit tangent bundle to S^2

  1. Nov 22, 2011 #1
    This is an example from Bott and Tu 's book DFAT(page 125).The example is in the image.I don't understand why can we get the local degree of the section s by constructing an vector field by parallel translation and calculate the rotating number of it.And why the local degree is 2?

    Could anybody recommend me a book on it?Thank you!:smile:
     

    Attached Files:

  2. jcsd
  3. Nov 22, 2011 #2

    lavinia

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    The local degree is the degree of the map of a small circle around the zero into itself. In this case the vector field has degree two - it winds twice around the circle.

    The local degree is the local degree associated with this section. Another section can have different local degrees around its zeros.
     
  4. Nov 22, 2011 #3
    Hi lavinia! Why it winds twice around the circle then the local degree is 2?Thank you.
     
  5. Nov 22, 2011 #4

    lavinia

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    The degree of a map from a compact manifold into another manifold is the number of points in the preimage of any regular value. This is a constant.

    For a map of degree two from a circle into another it would have to wrap around twice.

    But is just look at the picture of the vector field that you sent, you will see that the vector field wraps twice around small circle near infinity,
     
    Last edited: Nov 22, 2011
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