Smooth structure

Definition

A smooth structure on a manifold ${\displaystyle M}$ is a collection of smoothly equivalent smooth atlases. Here, a smooth atlas for a topological manifold ${\displaystyle M}$ is an atlas for ${\displaystyle M}$ such that each transition function is a smooth map, and two smooth atlases for ${\displaystyle M}$ are smoothly equivalent provided their union is again a smooth atlas for ${\displaystyle M.}$ This gives a natural equivalence relation on the set of smooth atlases.

A smooth manifold is a topological manifold ${\displaystyle M}$ together with a smooth structure on ${\displaystyle M.}$

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 smooth 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.

Equivalence of smooth structures

If ${\displaystyle \mu }$ and ${\displaystyle \nu }$ are two maximal atlases on ${\displaystyle M}$ the two smooth structures associated to ${\displaystyle \mu }$ and ${\displaystyle \nu }$ are said to be equivalent if there is a diffeomorphism ${\displaystyle f:M\to M}$ such that ${\displaystyle \mu \circ f=\nu .}$