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

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Discussion Overview

The discussion centers on the concept of the Euler class of the unit tangent bundle to the 2-sphere (S²), specifically exploring the local degree of a section and its relationship to vector fields and their winding numbers. The context includes theoretical aspects and mathematical reasoning related to differential geometry.

Discussion Character

  • Technical explanation
  • Mathematical reasoning

Main Points Raised

  • Some participants express confusion regarding how to determine the local degree of a section by constructing a vector field through parallel translation and calculating its winding number.
  • One participant states that the local degree is the degree of the map of a small circle around the zero into itself, asserting that the vector field has degree two as it winds twice around the circle.
  • Another participant clarifies that the degree of a map from a compact manifold into another is defined as the number of points in the preimage of any regular value, suggesting that a map of degree two must wrap around twice.
  • Participants reference a specific image of the vector field to illustrate the concept of winding around a small circle near infinity.

Areas of Agreement / Disagreement

The discussion reflects some agreement on the definition of local degree and its relation to winding numbers, but participants express uncertainty about the reasoning behind the local degree being two and how it is derived from the vector field's behavior.

Contextual Notes

Limitations include the reliance on visual representations (the image of the vector field) and the potential for varying local degrees associated with different sections around their zeros, which remains unresolved in the discussion.

kakarotyjn
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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:
 

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kakarotyjn said:
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:

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.
 
kakarotyjn said:
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,
 
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