# Confusion regarding compatible charts on smooth manifolds

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?

• Yes. When talking about charts of a differentiable manifold, they are implied to be taken from the differentiable structure. Sep 23, 2020 at 14:33

From now on, a “manifold” will mean a $$C^\infty$$-manifold. We use the terms “smooth” and $$C^\infty$$ interchangeably. [...] By a chart $$(U,\phi)$$ about $$p$$ in a manifold $$M$$, we will mean a chart in the differentiable structure of $$M$$ such that $$p \in U$$.
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.
Most likely you have seen a $$C^{\infty}$$ map defined along the lines "if there are charts such that...". This result helps bridge the gap between theory and practice: if you want to disprove a map is $$C^{\infty}$$, by the definition, you would have to check all charts (this is how you prove non-existence). However, after this result, you only have to find a pair of charts that don't make the local map be $$C^{\infty}$$.