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.
Read more about this topic: Smooth Structure
Famous quotes containing the word smooth:
“We were soon in the smooth water of the Quakish Lake,... and we had our first, but a partial view of Ktaadn, its summit veiled in clouds, like a dark isthmus in that quarter, connecting the heavens with the earth.”
—Henry David Thoreau (1817–1862)