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

[kakarotyjn]

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!

 

[lavinia]

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!

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.

 

[kakarotyjn]

Hi lavinia! Why it winds twice around the circle then the local degree is 2?Thank you.

 

[lavinia]

Hi lavinia! Why it winds twice around the circle then the local degree is 2?Thank you.

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,