Global diffeomorphism with tangent bundle

In summary: And for a 1-manifold, a single non-zero vector field always exists, so the tangent bundle is always trivial.
  • #1
Monocles
466
2
I am terribly confused on the issue of trivial tangent bundles. I understand intuitively why some tangent bundles are trivial and others are not, but I'm having trouble figuring out how to show it.

Even the most trivial example, show that [tex]T\mathbb{R}^n[/tex] is diffeomorphic to [tex]\mathbb{R}^{2n}[/tex] I am not seeing how to show. Showing that they are locally diffeomorphic is very easy, but every tangent bundle is locally diffeomorphic to the product space of the manifold with the appropriate Euclidean space. I am new to this topic so a geometrical route is preferred. For example, I know that if there exists a vector field with no zero vectors then the tangent bundle is trivial, but I don't know how to show why that is true, so that result does not help me.

Thanks!
 
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  • #2
[tex]
\mathbb{R}^n
[/tex]

has global coordinates. Use them to define natural global coordinates for the tangent bundle. Then notice that they define a global trivialization.
 
  • #3
Thanks, that gave me an idea. So, if I want to show that the tangent bundle on [tex]TS^1[/tex] is trivial, should I just find an atlas for the bundle, show that the coordinate charts are locally diffeomorphic to [tex]S^1 \times \mathbb{R}[/tex], and then show that the transition functions between the coordinate charts are smooth? I feel like that must still be missing a step somewhere, since the transition functions are smooth by definition.
 
  • #4
To prove that the tangent bundle of an n-dimensional manifold is trivial (and to find its trivialization) it is enough to find n vector fields that are linearly independent at every point. For the circle it is enough to find just one nowhere vanishing vector field (which should be easy).

Once you convince yourself that this is the case - it will help you in the future.
 
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  • #5
Monocles said:
I am terribly confused on the issue of trivial tangent bundles. I understand intuitively why some tangent bundles are trivial and others are not, but I'm having trouble figuring out how to show it.

Even the most trivial example, show that [tex]T\mathbb{R}^n[/tex] is diffeomorphic to [tex]\mathbb{R}^{2n}[/tex] I am not seeing how to show. Showing that they are locally diffeomorphic is very easy, but every tangent bundle is locally diffeomorphic to the product space of the manifold with the appropriate Euclidean space. I am new to this topic so a geometrical route is preferred. For example, I know that if there exists a vector field with no zero vectors then the tangent bundle is trivial, but I don't know how to show why that is true, so that result does not help me.

Thanks!

It is not true that a single vector field without zeros means that the tangent bundle is trivial. This is only true for oriented surfaces and for 1 manifolds like the circle. For instance the Klein bottle has a non-zero vector field but it's tangent bundle is not trivial.

I think that triviality formally means that there is a homeomorphism from the vector bundle into BxF ,where B is the base space of the bundle and F is a vector space, that is linear on each fiber and covers the identity map on B.
 
  • #6
lavinia said:
It is not true that a single vector field without zeros means that the tangent bundle is trivial.

Of course. You need, as I wrote, n linearly independent vector fields.

This is only true for oriented surfaces and for 1 manifolds like the circle. For instance the Klein bottle has a non-zero vector field but it's tangent bundle is not trivial.

Of course, because Klein's bottle is not 1-dimensional.
A simple and useful http://www.math.uchicago.edu/~womp/2001/vbex.pdf" :

Exercise 3. Show that an n-dimensional vector bundle E -> M is trivial if and only if there are n sections s1,..., sn which, in each fiber, are linearly independent. Show that all bundles have local systems of n linearly independent sections.​
 
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  • #7
arkajad said:
Of course. You need, as I wrote, n linearly independent vector fields.
Of course, because Klein's bottle is not 1-dimensional.
A simple and useful http://www.math.uchicago.edu/~womp/2001/vbex.pdf" :

Exercise 3. Show that an n-dimensional vector bundle E -> M is trivial if and only if there are n sections s1,..., sn which, in each fiber, are linearly independent. Show that all bundles have local systems of n linearly independent sections.​

I was just responding to the direct words of the writer and giving an example - not disagreeing with you or for that matter not not understanding the definition of trivial. Perhaps you can explain what you are getting at.

BTW: For an oriented surface (Riemannian manifold) it suffices to have a single non-zero vector field. The orientation provides the other.
 
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  • #8
lavinia said:
BTW: For an oriented surface (Riemannian manifold) it suffices to have a single non-zero vector field. The orientation provides the other.

Right.
 

Related to Global diffeomorphism with tangent bundle

1. What is a global diffeomorphism and how is it related to the tangent bundle?

A global diffeomorphism is a smooth, invertible mapping between two differentiable manifolds that preserves the smooth structure of the manifolds. In other words, it is a function that smoothly transforms one manifold into another. The tangent bundle, on the other hand, is a mathematical construction that assigns a tangent space to each point on a manifold. Global diffeomorphisms can be used to define a smooth mapping between tangent bundles, allowing for a consistent way to compare tangent spaces at different points on a manifold.

2. How is the concept of global diffeomorphism used in mathematics?

Global diffeomorphisms are used in a variety of mathematical fields, including differential geometry, topology, and dynamical systems. They are important because they allow for a smooth transformation of geometric objects, such as manifolds, and provide a way to compare these objects in a consistent manner. In particular, they are used in the study of smooth mappings between manifolds, which has applications in physics, engineering, and computer science.

3. What is the difference between a global diffeomorphism and a local diffeomorphism?

The main difference between a global diffeomorphism and a local diffeomorphism is the scope of their transformations. A global diffeomorphism is a smooth, invertible mapping between entire manifolds, while a local diffeomorphism is a smooth, invertible mapping between open subsets of a manifold. This means that a global diffeomorphism can transform the entire manifold, while a local diffeomorphism can only transform a small portion of it.

4. How are global diffeomorphisms related to symmetry?

Global diffeomorphisms are closely related to symmetry in mathematics. In particular, they can be seen as a type of symmetry that preserves the smooth structure of a manifold. Just as a symmetry in geometry is a transformation that leaves an object unchanged, a global diffeomorphism is a transformation that preserves the geometric properties of a manifold. This allows for a more general notion of symmetry in mathematics.

5. What are some real-world applications of global diffeomorphisms?

Global diffeomorphisms have a wide range of applications in the real world. In physics, they are used in the study of spacetime and general relativity, where they help to define smooth transformations between different coordinate systems. In computer graphics and animation, they are used to create smooth deformations of 3D models. They are also used in engineering and robotics, where they help to analyze and design smooth motions of mechanical systems.

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