Exercise 2.9 (Smoothness does not depend on the choice of charts II)

Question

Suppose that $F:M 	o N$ is a smooth map between smooth manifolds with or without boundary. Show that the coordinate representation of $F$ with respect to \emph{every} pair of smooth charts for $M$ and $N$ is smooth.

Elu Thingol · Accepted Answer

\begin{proof}
Let $(\widetilde{U},\widetilde{\varphi})$ and $(\widetilde{V},\widetilde{\psi})$ be smooth charts for $M$ and $N$ respectively and consider the set $\widetilde{U} \cap F^{-1}(\widetilde{V})$. If $\widetilde{U} \cap F^{-1}(\widetilde{V})$ is empty, the assertion follows vacuously, so assume that it is non-empty. By the smoothness of $F$, for each $p \in \widetilde{U} \cap F^{-1}(\widetilde{V})$, we can find a chart $(U_p,\varphi)$ containing $p$ and a chart $(V_p,\psi)$ containing $F(p)$ such that $F(U_p)\subseteq V_p$ and the coordinate representation
\begin{equation}
    \psi \circ F \circ \varphi^{-1}: \varphi(U_p) 	o \psi(V_p)
\end{equation}
is smooth. Furthermore, by smooth compatibility of $(\widetilde{U},\widetilde{\varphi})$ and $(U_p,\varphi)$, there is a smooth transition map
\begin{equation}
    \varphi \circ \widetilde{\varphi}^{-1}: \widetilde{\varphi}(\widetilde{U} \cap U_p) 	o \varphi(\widetilde{U} \cap U_p). 
\end{equation}

Similarly, there is a smooth transition map between $(\widetilde{V},\widetilde{\psi})$ and $(V,\psi)$:
\begin{equation}
    \widetilde{\psi} \circ \psi^{-1}: \psi(\widetilde{V} \cap V_p) 	o \widetilde{\psi}(\widetilde{V} \cap V_p). 
\end{equation}

Composing all three smooth maps into one, we obtain a smooth coordinate representation
\begin{align*}
    &(\widetilde{\psi} \circ \psi^{-1}) \circ (\psi \circ F \circ \varphi^{-1}) \circ (\varphi \circ \widetilde{\varphi}^{-1})\
    &= \widetilde{\psi} \circ F \circ \widetilde{\varphi}^{-1}: \widetilde{\varphi}(\widetilde{U} \cap U_p \cap F^{-1}(\widetilde{V})) 	o \widetilde{\psi}(\widetilde{V})
\end{align*}
Since our choice of $p \in \widetilde{U} \cap F^{-1}(\widetilde{V})$ was arbitrary, $F$ must be smooth with respect to $(\widetilde{U},\widetilde{\varphi})$ and $(\widetilde{V},\widetilde{\psi})$.
\end{proof}

Exercise 2.9 (Smoothness does not depend on the choice of charts II)

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Exercise 2.9 (Smoothness does not depend on the choice of charts II)

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