Smooth structure

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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.

Definition[edit]

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.

Maximal smooth atlases[edit]

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[edit]

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

Exotic spheres[edit]

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.

Related structures[edit]

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 structure by requiring the transition maps to be holomorphic.

References[edit]

  • Hirsch, Morris (1976). Differential Topology. Springer-Verlag. ISBN 3-540-90148-5.