Bundle and differential manifold

In summary, we discussed the concept of tangent spaces and tangent bundles on a differential manifold of dimension n. The tangent space is a vector space of dimension n that is in bijection with the derivation space, where each derivation is a directional derivative. The tangent bundle is then defined as the union of all tangent spaces at each point on the manifold. To demonstrate that the tangent bundle is a C^∞ manifold of dimension 2n, a topology is defined using charts that are compatible and open sets are defined using the projection map. The next step is to show that the projection map is a homeomorphism, which can be done by showing that its inverse is continuous and bijective. Finally, the differential of a C^∞ dif
  • #1
Calabi
140
2
Hello : let be a differential manifold [tex]C^{\infty}[/tex] : [tex]M[/tex] of dimension n.
I choose a point p.

In this point I can defined the tangent space. It's a vectoirial space of dimension n, I'll talk about it in a precedent thread, .
This space is in bijection with the derivation space : each derivation associated a real to each fonction [tex]C^{\infty}[/tex] define on a neighbourhood of p.
Each derivator is a directionnal derivate.

I defined the tangent bundle as : [tex]TM = \cup_{p \in M} (\{p\} \times T_{p}M)[/tex]. I want to demonstrate that this space is a differential [tex]C^{+\infty}[/tex] manifold of dimension 2n.

I beginn to define a topology on this space. Let [tex](U, \phi)[/tex] et [tex](V, \psi)[/tex] 2 charts, which are compatible [tex]C^{\infty}[/tex] and [tex]U \cap V \neq \varnothing[/tex]. I defined an open as [tex]\pi^{-1}(U) = \{ \{p\} \times T_{p}M / p \in U \}[/tex].

Like all the open define a topology which recover M, all the open I defined with my [tex]\pi^{-1}[/tex] on [tex]TM[/tex] defined a topology which recover [tex]TM[/tex].

Now I defined the same things on [tex](V, \psi)[/tex].

Now let's go back to [tex](U, \phi)[/tex] : I defined :
[tex] \Phi : \begin{pmatrix} \pi^{-1}(U) \rightarrow \phi(U) \times \mathbb{R}^{n} \subset \mathbb{R}^{2n} \\ (p, X_{p}) \rightarrow (\phi(p), d_{\phi(p)}(X_{p})) \end{pmatrix} [/tex].

I recall that [tex]d_{\phi(p)}[/tex] associated to each vectors of [tex]T_{p}M[/tex] a vectors from [tex]T_{\phi(p)}\mathbb{R}^{n}[/tex]. Like [tex]\forall x \in \mathbb{R}^{n}, T_{x}\mathbb{R}^{n} \simeq \mathbb{R}^{n}[/tex], I can identified [tex]d_{\phi(p)}(X_{p})[/tex] to an elements of [tex]\mathbb{R}^{n}[/tex]. With this natural components in the natural base.

I do the same things by defining [tex]\Psi[/tex].
And [tex](U, \phi)[/tex] et [tex](V, \psi)[/tex] are arbitrarlly choose.

So 2 question how to demonstrate that [tex]\Phi[/tex] is an homeomorphism please?

How to demosntrate that [tex]\Psi o \Phi^{-1}[/tex] is a [tex]C^{\infty}[/tex] diffeomorphisme please?

Thank you in advance and have a nice afternoon:biggrin:.
 
Physics news on Phys.org
  • #2
On ne doit qu'observer que la dérivée d'une fonction ##C^∞## est ##C^∞## elle-même( par définition de ##C^∞##) est que la dérivée d'une fonction de transition est un isomorphism lineare de l'espace tangente a chaque point.

Essayez de montrer que l'espace euclidien est une variété ##C^∞##.

Bonne journée.
 
  • #3
Salut Lavina. Merci à vous. First by the way : you could replace all the [tex]+\infty[/tex] by [tex]k[/tex].

I'm going to wright : [tex]\Psi o \Phi^{-1}(\phi(p), d_{\phi(p)}(X_{p}) )= (\psi o \phi^{-1}(p) , d_{\psi(p)} o d_{\phi(p)}^{-1}(X_{p}))[/tex] and I've got : [tex]\Phi(p, X_{p}) = (\phi(p), d_{\phi(p)}(X_{p})[/tex].
I know that [tex]\phi[/tex] like [tex]\psi[/tex] are homeomorphism. And that [tex]\psi o \phi^{-1}[/tex] is [tex]C^{+\infty}[/tex] because of the [tex]C^{+\infty}[/tex] differential manifold structure.

So for the first par of the 2 uplets it's done.

What about [tex]d_{\psi(p)} o d_{\phi(p)}^{-1}[/tex] is it an diffeormorphism [tex]C^{+\infty}[/tex] please?

And about [tex]d_{\phi(p)}[/tex], is it an homeomorphism please?

Thank you in advance and have a nice afternoon:biggrin:.
 
  • #4
Calabi said:
Salut Lavina. Merci à vous. First by the way : you could replace all the [tex]+\infty[/tex] by [tex]k[/tex].
No. The derivative of a k times differentiable function might only be (k-1) times differentiable. There are examples of continuously differentiable functions whose derivative is nowhere differentiable.

I'm going to wright : [tex]\Psi o \Phi^{-1}(\phi(p), d_{\phi(p)}(X_{p}) )= (\psi o \phi^{-1}(p) , d_{\psi(p)} o d_{\phi(p)}^{-1}(X_{p}))[/tex] and I've got : [tex]\Phi(p, X_{p}) = (\phi(p), d_{\phi(p)}(X_{p})[/tex].
I know that [tex]\phi[/tex] like [tex]\psi[/tex] are homeomorphism. And that [tex]\psi o \phi^{-1}[/tex] is [tex]C^{+\infty}[/tex] because of the [tex]C^{+\infty}[/tex] differential manifold structure.

So for the first par of the 2 uplets it's done.

What about [tex]d_{\psi(p)} o d_{\phi(p)}^{-1}[/tex] is it an diffeormorphism [tex]C^{+\infty}[/tex] please?

And about [tex]d_{\phi(p)}[/tex], is it an homeomorphism please?

Thank you in advance and have a nice afternoon:biggrin:.

As I said before, the differential of a ##C^∞## diffemomorphism is a ##C^∞## diffeomorphism.

- The differential of a smooth diffeomorphism is a diffeomorphism because it is smooth and invertible.
 
Last edited:
  • #5
Hello I wright for the probleme I demonstrate my charts is an homeomorphisme, it's continue and a bijection. The reverse fonction is also a continue bijection. I juste have to demonstrate now that my charts changing [tex]\Psi o \Phi^{-1}[/tex] is a [tex]C^{k-1}[/tex] diffeomorphism if my manifold is [tex]C^{k}[/tex]. How could I do please?

Thank you in advance and have a nice afternoon:biggrin:.
 
  • #6
lavinia said:
As I said before, the differential of a C∞C^∞ diffemomorphism is a C∞C^∞ diffeomorphism.

How to demonstrate it please? And

Thank you in advance and have a nice afternoon:biggrin:.
 
  • #7
Calabi said:
How to demonstrate it please? And

Thank you in advance and have a nice afternoon:biggrin:.

Calabi. Here are some questions that may help you.

- The differential of a diffeomorphism is a map between tangent spaces. This map is linear and is an isomorphism. Why?
- A smooth map is infinitely differentiable. What is the definition of infinitely differentiable for a multi-variable function?
- A smooth diffeomorphism by definition has a smooth inverse. What does this mean about the differential?
 
Last edited:

1. What is a bundle in differential manifold?

A bundle in differential manifold is a mathematical construct used to describe the relationship between a base space and a collection of fibers. The base space is typically a smooth manifold, and the fibers are typically vector spaces or manifolds themselves. Bundles are used in many areas of mathematics and physics, such as in the study of vector fields, differential equations, and gauge theories.

2. How is a bundle different from a manifold?

A manifold is a mathematical space that is locally Euclidean, meaning that it resembles a flat space in a small neighborhood around each point. A bundle, on the other hand, is a space that is made up of multiple manifolds, each of which is associated with a point in a base space. In other words, a bundle is a collection of manifolds that are "glued" together in a structured way.

3. What is the purpose of using bundles in differential manifold?

Bundles are used in differential manifold to study the relationship between different mathematical spaces, such as manifolds and vector spaces. They allow us to understand how these spaces are related and how they can be used to model real-world phenomena. Bundles also provide a way to generalize concepts from one space to another, making them a powerful tool in mathematics and physics.

4. Can you give an example of a bundle in differential manifold?

One example of a bundle in differential manifold is the tangent bundle of a smooth manifold. This bundle consists of a vector space (the tangent space) associated with each point in the base manifold. Another example is the Hopf bundle, which is used in topology to study the properties of spheres.

5. What are the applications of bundles in differential manifold?

Bundles have many applications in mathematics and physics. In differential geometry, they are used to study the geometry of manifolds and to define important concepts such as curvature and connections. In physics, bundles are used to describe the behavior of fields, such as electromagnetic and gravitational fields. They are also used in quantum mechanics to study the behavior of particles and their interactions.

Similar threads

  • Differential Geometry
Replies
10
Views
639
  • Differential Geometry
Replies
2
Views
511
  • Differential Geometry
Replies
20
Views
2K
  • Differential Geometry
Replies
9
Views
404
Replies
17
Views
3K
  • Differential Geometry
Replies
13
Views
2K
Replies
4
Views
1K
  • Differential Geometry
Replies
3
Views
1K
Replies
4
Views
2K
  • Differential Geometry
Replies
10
Views
2K
Back
Top