In
mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, 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 it has over some domain, called ''differentiability class''. At the very minimum, a function could be considered smooth if ...
. In particular, a smooth structure allows one to perform
mathematical analysis
Analysis is the branch of mathematics dealing with continuous functions, limits, and related theories, such as differentiation, integration, measure, infinite sequences, series, and analytic functions.
These theories are usually studied ...
on the manifold.
Definition
A smooth structure on a manifold
is a collection of smoothly equivalent smooth atlases. Here, a smooth atlas for a topological manifold
is an
atlas
An atlas is a collection of maps; it is typically a bundle of maps of Earth or of a region of Earth.
Atlases have traditionally been bound into book form, but today many atlases are in multimedia formats. In addition to presenting geogra ...
for
such that each
transition function is a
smooth map, and two smooth atlases for
are smoothly equivalent provided their
union is again a smooth atlas for
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.
Each equivalence relatio ...
on the set of smooth atlases.
A
smooth manifold is a topological manifold
together with a smooth structure on
Maximal smooth atlases
By taking the union of all
atlases
An atlas is a collection of maps; it is typically a bundle of maps of Earth or of a region of Earth.
Atlases have traditionally been bound into book form, but today many atlases are in multimedia formats. In addition to presenting geograph ...
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
compact
Compact as used in politics may refer broadly to a pact or treaty; in more specific cases it may refer to:
* Interstate compact
* Blood compact, an ancient ritual of the Philippines
* Compact government, a type of colonial rule utilized in Britis ...
, then one can find an atlas with only finitely many charts.
Equivalence of smooth structures
Let
and
be two maximal atlases on
The two smooth structures associated to
and
are said to be equivalent if there is a
diffeomorphism such that
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
In an area of mathematics called differential topology, an exotic sphere is a differentiable manifold ''M'' that is homeomorphic but not diffeomorphic to the standard Euclidean ''n''-sphere. That is, ''M'' is a sphere from the point of view of a ...
.
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
-times continuously differentiable; or strengthened, so that we require the transition maps to be real-analytic. Accordingly, this gives a
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
*
*
References
*
*
*
{{Manifolds
Differential topology
Structures on manifolds