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

## Main Question or Discussion Point

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! #### Attachments

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lavinia
Gold Member
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.

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

lavinia