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

In summary, the local degree is the degree of the map of a small circle around the zero into itself, which in this case is 2 because the vector field winds twice around the circle. It is associated with a section and can vary for different sections. For a map of degree two from a circle into another, it would have to wrap around twice. The vector field wraps twice around a small circle near infinity but only once around a small circle near the zero.
  • #1
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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  • #2
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.
 
  • #3
Hi lavinia! Why it winds twice around the circle then the local degree is 2?Thank you.
 
  • #4
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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  • #5


The Euler class is a fundamental concept in differential geometry and topology, and its calculation in this example from Bott and Tu's book DFAT is a great illustration of its significance. The unit tangent bundle to S^2 is the set of all unit tangent vectors to the two-dimensional sphere, and the Euler class is a characteristic class that measures the obstructions to orienting this bundle.

In this example, the authors construct a vector field on the unit tangent bundle by parallel translation. This means that they take a tangent vector at one point on the sphere and move it along a geodesic curve, keeping its direction constant. By doing this, they create a vector field that is tangent to the sphere at every point, and thus defines a section of the tangent bundle.

The local degree of this section is then calculated by counting the number of times the vector field rotates around the sphere as it moves along a closed curve. This is known as the rotating number, and it is a topological invariant that reflects the underlying geometry of the sphere. In this case, the local degree is found to be 2, which means that the vector field rotates around the sphere twice as it moves along a closed curve.

This result is significant because it tells us that the unit tangent bundle to S^2 is non-orientable, meaning that it cannot be given a consistent orientation. This is reflected in the fact that the Euler class of this bundle is non-zero, indicating the presence of obstructions to orientability.

As for further reading, I would recommend "Topology from the Differentiable Viewpoint" by John Milnor, which covers the fundamentals of differential topology and includes a chapter on characteristic classes. Additionally, "Differential Forms in Algebraic Topology" by Raoul Bott and Loring Tu is a great resource for understanding the Euler class and other characteristic classes.
 

What is the Euler class of the unit tangent bundle to S^2?

The Euler class of the unit tangent bundle to S^2 is a topological invariant that measures the obstruction to constructing a nowhere-vanishing section of the bundle. It is a characteristic class that can be computed using the Pontryagin square of the first Chern class of the tangent bundle.

How is the Euler class of the unit tangent bundle to S^2 related to the topology of S^2?

The Euler class of the unit tangent bundle to S^2 is related to the topology of S^2 through the Gauss-Bonnet theorem. This theorem states that the integral of the Euler class over the surface is equal to the Euler characteristic of the surface, which is a topological invariant.

Can the Euler class of the unit tangent bundle to S^2 be used to classify differentiable structures on S^2?

Yes, the Euler class of the unit tangent bundle to S^2 can be used to distinguish between different differentiable structures on S^2. In particular, the Euler class detects the non-orientability of a surface and thus can distinguish between orientable and non-orientable differentiable structures.

How is the Euler class of the unit tangent bundle to S^2 related to the curvature of S^2?

The Euler class of the unit tangent bundle to S^2 is related to the curvature of S^2 through the Gauss-Bonnet theorem. This theorem states that the integral of the Euler class over the surface is equal to the total Gaussian curvature of the surface, which is a measure of its global curvature.

Can the Euler class of the unit tangent bundle to S^2 be generalized to other manifolds?

Yes, the Euler class of the unit tangent bundle can be generalized to other manifolds. In fact, for any compact, oriented manifold, there exists a similar topological invariant called the Euler characteristic, which is defined in terms of the Euler class of the tangent bundle. This concept can also be extended to higher dimensions, where there are higher Euler classes that measure different aspects of the curvature of a manifold.

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