Definition:Compatible Atlases

Definition
Let $M$ be a locally Euclidean space of dimension $d$.

Let $\mathscr F, \mathscr G$ be atlases of class $C^k$ on $M$.

Then $\mathscr F$ and $\mathscr G$ are compatible iff for all charts $\left({U, \phi}\right) \in \mathscr F$ and $\left({V, \psi}\right) \in \mathscr G$:


 * $\phi \circ \psi^{-1}: \psi \left({U \cap V}\right) \to \phi \left({U \cap V}\right)$

is of class $C^k$.