- #1

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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.