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:
“Less smooth than her Skin and less white than her breast
Was this pollisht stone beneath which she lyes prest
Stop, Reader, and Sigh while thou thinkst on the rest
With a just trim of Virtue her Soul was endud
Not affectedly Pious nor secretly lewd,
She cut even between the Cocquet and the Prude.”
—Matthew Prior (16641721)