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:
“A leaf that is supposed to grow is full of wrinkles and creases before it develops; if one doesn’t have the patience and wants the leaf to be as smooth as a willow leaf from the start, then there is a problem.”
—Johann Wolfgang Von Goethe (1749–1832)