Smooth structure

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 ${\displaystyle M.}$

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.

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.

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.

• Smooth frame – Generalization of an ordered basis of a vector spacePages displaying short descriptions of redirect targets
• Atlas (topology) – Set of charts that describes a manifold