Bundle and differential manifold

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Main Question or Discussion Point

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 wanna 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 lets 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:.
 

Answers and Replies

  • #2
lavinia
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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
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Salut Lavina. Merci à vous. First by the way : you could replace all the [tex]+\infty[/tex] by [tex]k[/tex].

I'm gonna 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
lavinia
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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 gonna 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.
 
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  • #5
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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
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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
lavinia
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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?
 
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