Simple question about definition of tangent bundle

In summary, Nakahara suggests that the definition of a bundle over M is a continuous map from topological spaces to M, and that the tangent bundle is the only possible bundle if the definition is used liberally.
  • #1
pellman
684
5
So I'm trying to learn about fibre bundles and I am looking at the example of a tangent bundle.

Given a differentiable manifold M. Denote the tangent space at [tex]p \in M[/tex] by [tex]T_p M[/tex]. Is the definition of the tangent bundle

[tex]TM = \lbrace (p, T_p M)|p \in M \rbrace[/tex]

or is it

[tex]TM = \lbrace (p, V)|p \in M , V \in T_p M\rbrace[/tex]?


Maybe I'm splitting hairs but there should be standard definition of one or the other, right?

I can discuss further why I think it matters but first let's just see if anyone is certain about the answer.
 
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  • #2
pellman said:
Is the definition of the tangent bundle
I feel I should point out that the definition of a bundle over M is a continuous map of topological spaces with codomain M. In other words, you need to specify:

1. A topological space E, which consists of
1a. A set of points |E|
1b. A topology on |E|
2. A continuous function E --> M (often called the 'projection map', or the 'structure map')
[tex]TM = \lbrace (p, T_p M)|p \in M \rbrace[/tex]
Assuming you use the obvious projection map, this is a very boring bundle: the projection is bijective! And if you include the local triviality condition, the projection is actually a homeomorphism!

[tex]TM = \lbrace (p, V)|p \in M , V \in T_p M\rbrace[/tex]?
Assuming you use the obvious projection map and choose the appropriate topology, this is indeed a tangent bundle. (There are many tangent bundles; they're just all isomorphic)
 
  • #3
Hurkyl, you da man. Thanks for the quick response.

So bijective is bad? That's part of what I don't get. I'm following Nakahara. You can see the page I am on here http://books.google.com/books?id=cH...&hl=en&sa=X&oi=book_result&resnum=1&ct=result

So when he says [tex]\pi^{-1}(p)=T_p M[/tex] he's being very loose with the inverse notation, right? [tex]\pi^{-1}[/tex] doesn't really exist, since [tex]\pi((p,V))=p[/tex] for every [tex]V \in T_p M[/tex]?
 
  • #4
pellman said:
So bijective is bad? That's part of what I don't get.
It would be -- roughly speaking such a bundle has only one section. If it were the tangent bundle, that would mean that there is exactly one vector field.

So when he says [tex]\pi^{-1}(p)=T_p M[/tex]
He's using the "inverse image" function, and being (very slightly) liberal with equality, since with the definition you gave, the fiber should be [itex]\{ p \} \times T_p M[/itex].
 
  • #5
Ok. That gives me enough to press on. I'm sure I will get it when I see other examples. Thanks again.
 

What is the definition of the tangent bundle?

The tangent bundle is a mathematical concept used in differential geometry to describe the collection of all tangent spaces of a smooth manifold. It is a vector bundle that associates each point on the manifold with a tangent space, which is a vector space that contains all possible directions in which a curve can pass through the point.

What is the purpose of the tangent bundle?

The tangent bundle is important because it allows us to study the behavior of curves on a manifold. By associating each point with a tangent space, we can examine how curves change as they move along the manifold. This is useful in many areas of mathematics, including differential equations and optimization.

How is the tangent bundle related to the cotangent bundle?

The cotangent bundle is the dual space of the tangent bundle. This means that while the tangent bundle describes the possible directions of curves at a point, the cotangent bundle describes the possible directions of differential forms at the same point. They are closely related and often used together in differential geometry.

Can the tangent bundle be visualized?

Yes, the tangent bundle can be visualized in some cases. For example, if the manifold is a surface in three-dimensional space, the tangent bundle at each point on the surface can be visualized as a plane tangent to the surface. However, in higher dimensions, it becomes more difficult to visualize the tangent bundle.

What is the difference between the tangent bundle and the tangent space?

The tangent space is the vector space associated with a single point on a manifold, while the tangent bundle is the collection of all tangent spaces at each point on the manifold. The tangent bundle is a vector bundle, meaning it is a collection of vector spaces that vary smoothly over the manifold. The tangent space, on the other hand, is simply a vector space associated with a single point.

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