Smooth structure

In mathematics, a smooth structure on a manifold allows for an unambiguous notion of smooth function. In particular, a smooth structure allows one to perform mathematical analysis on the manifold.

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

A smooth manifold is a topological manifold M together with a smooth structure on M.

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.

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

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