Smooth compatibility of charts and atlases

Definition

Two coordinate charts $(U,\varphi)$ and $(V,\psi)$ on the same topological manifold are smoothly compatible if either $U\cap V$ is empty or the coordinate transition map

\psi\circ\varphi^{-1}:\varphi(U\cap V)\longrightarrow\psi(U\cap V)

is smooth with smooth inverse. Two smooth atlases are compatible if every chart in one is smoothly compatible with every chart in the other; equivalently, their union is a smooth atlas.

Equivalent formulations

Because the two transition maps are inverses, compatibility can equivalently be stated by requiring $\psi\circ\varphi^{-1}$ to be a diffeomorphism. Compatibility of smooth atlases is an equivalence relation, and two smooth atlases are compatible exactly when they generate the same maximal smooth atlas.

Scope of the terminology

Compatibility depends on the regularity category. Smooth atlases require smooth coordinate changes, while $C^k$ , real-analytic, complex, and other atlases impose their corresponding transition-map conditions. Thus the bare phrase “compatible atlases” should be interpreted only after the category has been specified.

References

John M. Lee, Introduction to Smooth Manifolds, 2nd ed., Springer, 2012. DOI record. Relevant: smooth structures and smoothly compatible charts.