Bundle and differential manifold

[Calabi]

Hello : let be a differential manifold C^{\infty} : M 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 C^{\infty} define on a neighbourhood of p.
Each derivator is a directionnal derivate.

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

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

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

Now I defined the same things on (V, \psi).

Now let's go back to (U, \phi) : I defined :
\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}.

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

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

So 2 question how to demonstrate that \Phi is an homeomorphism please?

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

Thank you in advance and have a nice afternoon.

 

[lavinia]

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.

 

[Calabi]

Salut Lavina. Merci à vous. First by the way : you could replace all the +\infty by k.

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

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

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

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

Thank you in advance and have a nice afternoon.

 

[lavinia]

Salut Lavina. Merci à vous. First by the way : you could replace all the +\infty by k.

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

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

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

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

Thank you in advance and have a nice afternoon.

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.

 

[Calabi]

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 \Psi o \Phi^{-1} is a C^{k-1} diffeomorphism if my manifold is C^{k}. How could I do please?

Thank you in advance and have a nice afternoon.

 

[Calabi]

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.

 

[lavinia]

How to demonstrate it please? And

Thank you in advance and have a nice afternoon.

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?