Let $M_1$, $M_2$, and N be manifolds of dimensions $m_1$, $m_2$, and $n$ respectively. Prove that a map $(f_1,f_2)$: $N$ -> $M_1\times M_2$ is smooth ($C^\infty$) if and only if $f_i$: $N$ ->$M_i$ i=1,2 are both smooth.
Attempted answer in the -> direction:
Let $p\in N$ and $(U,\phi)$ be a chart around $p$ in $N$. Let $(V,\psi)$ be a chart around of $(f_1,f_2)(p)$ in $M_1\times M_2$ and $(V_i,\psi_i)$ be a chart around of $f_i(p)$ in $M_i$. We know $\pi_i \circ \psi \circ (f_1,f_2) \circ \phi^{-1}$ is smooth because the projection is smooth and by hypothesis $\psi \circ (f_1,f_2) \circ \phi^{-1}$ is smooth.
(This is the part that doesn't convince me)
$\pi_i \circ \psi \circ (f_1,f_2) \circ \phi^{-1} = \psi_i \circ \pi_i \circ (f_1,f_2) \circ \phi^{-1}$. Therefore, $\pi_i \circ (f_1,f_2) = f_i$ is smooth.
I would like to know if this attempted answer in the -> direction is correct and how does the other direction goes.
Best Answer
The idea is following.
If the map $f_1\times f_2:N\to M_1\times M_2$ is smooth,you just notice that $f_i=\pi_i\circ (f_1\times f_2)$ which is smooth since $\pi_i$ is smooth.(The composition of smooth map also smooth,you should to know and prove which is easy by definition).
For the other side,if $f_i$ are smooth.For each $p\in N$ and $f(p)=(p_1,p_2)\in M_1\times M_2$,there exists the local coordinate $(U_1\times U_2,\varphi_1\times \varphi_2)$ where $(U_1,\varphi_1)$ and $(U_2,\varphi_2)$ are the local coordinates of $M_1$ and $M_2$ respectively s.t. $p_1\in M_1,p_2\in M_2$ and the local coordinate $(V,\psi)$ of $N$ s.t. $p\in V$.In this local coordinate we have $$\varphi_i\circ f_i\circ \psi^{-1}:\psi(V)\to \varphi_i(U_i)$$ are smooth(you should think how can we take $V$ by definition).Then we have $$(\varphi_1\times \varphi_2)\circ (f_1\times f_2)\circ \psi^{-1}:\psi(V)\to \varphi_1\times \varphi_2(U_1\times U_2)$$ s.t. $(\varphi_1\times \varphi_2)\circ (f_1\times f_2)\circ \psi^{-1}(x)=(\varphi_1\circ {f_1}\circ \psi^{-1}(x),\varphi_2\circ{f_2}\circ\psi^{-1}(x))$ is smooth.Since for all $p\in N$ hold,then the proposion has proved.