mathematics Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...

, a smooth structure on a

manifold In mathematics, a manifold is a topological space that locally resembles Euclidean space near each point. More precisely, an n-dimensional manifold, or ''n-manifold'' for short, is a topological space with the property that each point has a N ...

allows for an unambiguous notion of

smooth function In mathematical analysis, the smoothness of a function is a property measured by the number of continuous derivatives (''differentiability class)'' it has over its domain. A function of class C^k is a function of smoothness at least ; t ...

. In particular, a smooth structure allows

mathematical analysis Analysis is the branch of mathematics dealing with continuous functions, limit (mathematics), limits, and related theories, such as Derivative, differentiation, Integral, integration, measure (mathematics), measure, infinite sequences, series ( ...

to be performed on the manifold.

Definition

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 An atlas is a collection of maps; it is typically a bundle of world map, maps of Earth or of a continent or region of Earth. Advances in astronomy have also resulted in atlases of the celestial sphere or of other planets. Atlases have traditio ...

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 In mathematics, an equivalence relation is a binary relation that is reflexive, symmetric, and transitive. The equipollence relation between line segments in geometry is a common example of an equivalence relation. A simpler example is equ ...

on the set of smooth atlases. A

smooth manifold In mathematics, a differentiable manifold (also differential manifold) is a type of manifold that is locally similar enough to a vector space to allow one to apply calculus. Any manifold can be described by a collection of charts (atlas). One may ...

is a topological manifold

M

together with a smooth structure on

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 smooth 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

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, then one can find an atlas with only finitely many charts.

Equivalence of smooth structures

\mu

and

\nu

are two maximal atlases on

M

the two smooth structures associated to

\mu

and

\nu

are said to be equivalent if there is a

diffeomorphism In mathematics, a diffeomorphism is an isomorphism of differentiable manifolds. It is an invertible function that maps one differentiable manifold to another such that both the function and its inverse are continuously differentiable. Definit ...

f : M \to M

such that

\mu \circ f = \nu.

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 the transition maps are only required to be

k

-times continuously differentiable; or strengthened, so that the transition maps are required to be real-analytic. Accordingly, this gives a

C^k

or (real-)analytic structure on the manifold rather than a smooth one. Similarly, a complex structure can be defined by requiring the transition maps to be holomorphic.

References

* * * {{Manifolds Differential topology Structures on manifolds