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

  • Thread starter kakarotyjn
  • Start date
  • #1
98
0
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:
 

Attachments

  • Bott page125 1.jpg
    Bott page125 1.jpg
    35.5 KB · Views: 474
  • Bott page125 2.jpg
    Bott page125 2.jpg
    17.3 KB · Views: 424

Answers and Replies

  • #2
lavinia
Science Advisor
Gold Member
3,246
629
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.
 
  • #3
98
0
Hi lavinia! Why it winds twice around the circle then the local degree is 2?Thank you.
 
  • #4
lavinia
Science Advisor
Gold Member
3,246
629
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,
 
Last edited:

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

Replies
6
Views
7K
  • Last Post
Replies
8
Views
11K
  • Last Post
Replies
3
Views
2K
  • Last Post
Replies
6
Views
2K
  • Last Post
Replies
2
Views
1K
  • Last Post
Replies
2
Views
2K
Replies
11
Views
2K
Replies
55
Views
5K
Replies
17
Views
1K
  • Last Post
Replies
4
Views
2K
Top