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.

Smooth atlases

An atlas $\{(U_{\alpha },\varphi _{\alpha })\}$ for a topological manifold M is said to be smooth iff each transition function is a smooth map.

Two atlases for M are smoothly equivalent provided their union is a smooth atlas for M. This gives a natural equivalence relation on the set of smooth atlases.

Formal definition of smooth structure

A smooth structure on a manifold is a collection of smoothly equivalent smooth atlases.

Exotic spheres

John Milnor showed in 1956 that the 7-dimensional sphere admits a smooth structure that is distinct from the standard smooth structure. A sphere equipped with a nonstandard smooth structure is called an exotic sphere.

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.

Related structures

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

References

• Lee, John M. (2006). Introduction to Smooth Manifolds. Springer-Verlag. ISBN 978-0-387-95448-6.
• Sepanski, Mark R. (2007). Compact Lie Groups. Springer-Verlag. ISBN 978-0-387-30263-8.