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I think I found a map but I cannot prove that is a diffeomorphism. That is the map :
$F : T(M \times N) \to T(M) \times T(N)$ defined by $F((m,n),X) = ((m,(\pi_{1})_{\star,(m,n)}(X)),(n,(\pi_{2})_{\star,(m,n)}(X)))$
where $\pi_1 : M \times N \to M$ and $ \pi_2 : M \times N \to N$ are the projections and the "little star" represent the push-forward.

My try:

Now the idea using local charts I think is the following :

Let $(U,\phi)$ be a local chart for $M$ , $(V,\psi)$ be a local chart for $N$, and so $(U\times V,\Phi:= \phi\times \psi)$ is a local chart for $M\times N$ by definition, and $(TU, \tilde \phi)$ $(TV,\tilde \psi)$ $(T(U\times V), \tilde\Phi)$ will be the induced local charts in $TM, TN $ and $T(M\times N)$. Now the only problem is to check that this maps are in fact smooth. Now I will do this for $F$.

So in these local charts we have that $(\tilde \phi\times \tilde \psi)\circ F\circ(\tilde \Phi)^{-1}(x_1,...x_m,y_1,...,y_n,\epsilon_1,...\epsilon_m,\eta_1,...,\eta_n)=\\= (\tilde \phi\times \tilde \psi)\circ F(\phi^{-1}(x_1,...x_m),\psi^{-1}(y_1,...,y_n),\epsilon_1,...,\epsilon_m,\eta_1,...,\eta_n)=\\=(\tilde\phi\times \tilde \psi)(\phi^{-1}(x_1,...x_m),\epsilon_1,...,\epsilon_m,\psi^{-1}(y_1,...,y_n),\eta_1,...,\eta_n)=\\ =(x_1,...,x_m,\epsilon_1,...\epsilon_m,y_1,...,y_n,\eta_1,...,\eta_n)$

Now to clarify what is happening when we apply $F$ at the tangent vector that this $v=\epsilon_1 \frac{\partial}{\partial x_1}+...+\epsilon_n \frac{\partial}{\partial x_n}+...+\eta_1 \frac{\partial}{\partial y_1}+...+\eta_n \frac{\partial}{\partial y_n}$, we need to look at the diferentials $d\pi_1$ and $d\pi_2$. I will only look at one of them since the other is analogous. So for example at $d\pi_1(\frac{\partial}{\partial x_i})=\sum_{j=1}^{m}\frac{(\phi\circ \pi_1 \circ (\phi\times \psi)^{-1} )^j}{\partial x_i}\frac{\partial }{\partial x_j}=\frac{\partial}{\partial x_i}$, and $d\pi_1(\frac{\partial}{\partial y_i})=\sum_{j=1}^{m}\frac{(\psi\circ \pi_1 \circ (\phi\times \psi)^{-1} )^j}{\partial y_i}\frac{\partial }{\partial x_j}=0$.

My question : How could I construct $F^{-1}$? could anyone help?

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  • $\begingroup$ This math.stackexchange.com/questions/3109967/… may give you an answer. $\endgroup$ Dec 9, 2020 at 16:23
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    $\begingroup$ It does defenitely differ from my question $\endgroup$ Dec 9, 2020 at 16:29
  • $\begingroup$ @Reza Yes it does differ, but with some effort it answers your question. This shows that they are isomorphic as vector bundles over $M\times N$, thus as smooth manifolds. $\endgroup$
    – Didier
    Dec 9, 2020 at 18:38

1 Answer 1

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Adding to @Bargabbiati's answer. In proposition 3.14, proved in problem 3-2 of this same book, it is proven that \begin{equation*} \alpha (v) = (d {\pi _{1}} _{p} (v), ..., d {\pi _k}_p (v)) \end{equation*} for all $v \in T_p (M_1 \times \cdots \times M_k)$ is an isomorphism between this tangent space and the direct sum $$\bigoplus T_{p_i} M_i.$$

That being said, such $F$, as you defined it, is an isomorphism (that is almost clear).

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