Smooth Structure - Maximal Smooth Atlases

Maximal Smooth Atlases

By taking the union of all atlases belonging to a smooth structure, we obtain a maximal smooth atlas. This atlas contains every chart that is compatible with the smooth structure. There is a natural one to one correspondence between smooth structures and maximal smooth atlases. Thus, we may regard a smooth structure as a maximal atlas and vice versa.

In general, computations with the maximal atlas of a manifold are rather unwieldy. For most applications, it suffices to choose a smaller atlas. For example, if the manifold is compact, then one can find an atlas with only finitely many charts.