Let $(U, \phi), (V, \psi)$ two charts of the same manifold. We say that two charts are $C^\infty$-compatible if the following two maps
$$\phi \circ \psi^{-1} : \psi(U \cap V) \to \phi(U \cap V), \; \psi \circ \phi^{-1} : \phi(U \cap V) \to \psi(U \cap V)$$
are $C^\infty$.
Can someone explain me what's the intuitive meaning of charts compatibility?
Best Answer
The essential fact: in the intersection of the domains $U\cap V$ both charts will give the same answer to the question "is $f:M\longrightarrow\Bbb R$ differentiable at $x_0\in U\cap V$?"