I am reading Loring Tu's "Introduction to Manifolds" and I came across the following proposition:
Suppose $F:N \to M$ is $C^\infty$ at $p \in N$. If $(U, \phi)$ is any chart about $p$ in $N$ and $(V, \psi)$ is any chart about $F(p)$ in $M$, then $\psi \circ F \circ \phi^{-1}$ is $C^\infty$ at $\phi(p)$.
Proof. Since $F$ is $C^\infty$ at $p \in N$, there are charts $(U_\alpha, \phi_\alpha)$ about $p$ in $N$ and $(V_\beta, \psi_\beta)$ about $F(p)$ in $M$ such that $\psi_\beta \circ F \circ \phi_\alpha^{-1}$ is $C^\infty$ at $\phi_\alpha(p)$. By the $C^\infty$ compatibility of charts in a differentiable structure, both $\phi_\alpha \circ \phi$ and $\psi \circ \psi_\beta^{-1}$ on open subset of Euclidean spaces. Hence, the composite
$$
\psi \circ F \circ \phi^{-1} = (\psi \circ \psi_\beta^{-1}) \circ (\psi_\beta \circ F \circ \phi_\alpha^{-1}) \circ (\phi_\alpha \circ \phi^{-1})
$$
is $C^\infty$ at $\phi(p)$.
What I don't understand is the reason $\phi$ and $\phi_\alpha$ (and also $\psi$ and $\psi_\beta$) should be compatible. Are all charts on a smooth manifold compatible? Or does the author mean any chart in the differentiable structure by the expression any chart?
Best Answer
Yes. Quotation from the end of section 5.3 (p. 53):
This means that the charts occurring in Definition 6.5 and Proposition 6.7 are tacitly assumed to belong to the fixed differentiable structure which determines $M$ as a smooth manifold. In particular, the charts $(U,\phi)$ and $(U_\alpha,\phi_\alpha)$ as well as the charts $(V,\psi)$ and $(V_\beta,\psi_\beta)$ are automatically compatible.
Note that the same applies for Definition 6.1 and Remark 6.2. See Smoothness of a function is independent from the chart .