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Bundle and differential manifold

  1. Jan 28, 2015 #1
    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:.
     
  2. jcsd
  3. Jan 28, 2015 #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.
     
  4. Jan 28, 2015 #3
    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:.
     
  5. Jan 29, 2015 #4

    lavinia

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    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.

    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: Jan 29, 2015
  6. Feb 8, 2015 #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:.
     
  7. Feb 8, 2015 #6
    How to demonstrate it please? And

    Thank you in advance and have a nice afternoon:biggrin:.
     
  8. Feb 8, 2015 #7

    lavinia

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    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: Feb 8, 2015
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