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.^[1]

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 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[]

Let ${\displaystyle \mu }$ and ${\displaystyle \nu }$ be 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 ${\displaystyle \mu \circ f=\nu .}$^{[citation needed]}

Exotic spheres[]

John Milnor showed in 1956 that the 7-dimensional sphere admits a smooth structure that is not equivalent to the standard smooth structure. A sphere equipped with a nonstandard smooth structure is called an exotic sphere.

E8 manifold[]

The E8 manifold is an example of a topological manifold that does not admit a smooth structure. This essentially demonstrates that Rokhlin's theorem holds only for smooth structures, and not topological manifolds in general.

Related structures[]

The smoothness requirements on the transition functions can be weakened, so that we only require the transition maps to be ${\displaystyle k}$-times continuously differentiable; or strengthened, so that we require the transition maps to be real-analytic. Accordingly, this gives a ${\displaystyle C^{k}}$ or (real-)analytic structure on the manifold rather than a smooth one. Similarly, we can define a complex structure by requiring the transition maps to be holomorphic.

See also[]

• Smooth frame
• Atlas (topology) – Set of charts that describes a manifold

References[]