atlas (mathematics)
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In mathematics, particularly
topology In mathematics, topology (from the Greek words , and ) is concerned with the properties of a geometric object that are preserved under continuous deformations, such as stretching, twisting, crumpling, and bending; that is, without closing ...
, one describes a manifold using an atlas. An atlas consists of individual ''charts'' that, roughly speaking, describe individual regions of the manifold. If the manifold is the surface of the Earth, then an atlas has its more common meaning. In general, the notion of atlas underlies the formal definition of a manifold and related structures such as
vector bundle In mathematics, a vector bundle is a topological construction that makes precise the idea of a family of vector spaces parameterized by another space X (for example X could be a topological space, a manifold, or an algebraic variety): to every p ...
s and other fiber bundles.


Charts

The definition of an atlas depends on the notion of a ''chart''. A chart for a
topological space In mathematics, a topological space is, roughly speaking, a geometrical space in which closeness is defined but cannot necessarily be measured by a numeric distance. More specifically, a topological space is a set whose elements are called po ...
''M'' (also called a coordinate chart, coordinate patch, coordinate map, or local frame) is a
homeomorphism In the mathematical field of topology, a homeomorphism, topological isomorphism, or bicontinuous function is a bijective and continuous function between topological spaces that has a continuous inverse function. Homeomorphisms are the isomor ...
\varphi from an open subset ''U'' of ''M'' to an open subset of a
Euclidean space Euclidean space is the fundamental space of geometry, intended to represent physical space. Originally, that is, in Euclid's ''Elements'', it was the three-dimensional space of Euclidean geometry, but in modern mathematics there are Euclidean ...
. The chart is traditionally recorded as the ordered pair (U, \varphi).


Formal definition of atlas

An atlas for a
topological space In mathematics, a topological space is, roughly speaking, a geometrical space in which closeness is defined but cannot necessarily be measured by a numeric distance. More specifically, a topological space is a set whose elements are called po ...
M is an
indexed family In mathematics, a family, or indexed family, is informally a collection of objects, each associated with an index from some index set. For example, a ''family of real numbers, indexed by the set of integers'' is a collection of real numbers, wher ...
\ of charts on M which covers M (that is, \bigcup_ U_ = M). If the
codomain In mathematics, the codomain or set of destination of a function is the set into which all of the output of the function is constrained to fall. It is the set in the notation . The term range is sometimes ambiguously used to refer to either th ...
of each chart is the ''n''-dimensional
Euclidean space Euclidean space is the fundamental space of geometry, intended to represent physical space. Originally, that is, in Euclid's ''Elements'', it was the three-dimensional space of Euclidean geometry, but in modern mathematics there are Euclidean ...
, then M is said to be an ''n''-dimensional manifold. The plural of atlas is ''atlases'', although some authors use ''atlantes''. An atlas \left( U_i, \varphi_i \right)_ on an n-dimensional manifold M is called an adequate atlas if the image of each chart is either \R^n or \R_+^n, \left( U_i \right)_ is a locally finite open cover of M, and M = \bigcup_ \varphi_i^\left( B_1 \right), where B_1 is the open ball of radius 1 centered at the origin and \R_+^n is the closed half space. Every second-countable manifold admits an adequate atlas. Moreover, if \mathcal = \left( V_j \right)_ is an open covering of the second-countable manifold M then there is an adequate atlas \left( U_i, \varphi_i \right)_ on M such that \left( U_i\right)_ is a refinement of \mathcal.


Transition maps

A transition map provides a way of comparing two charts of an atlas. To make this comparison, we consider the composition of one chart with the inverse of the other. This composition is not well-defined unless we restrict both charts to the intersection of their domains of definition. (For example, if we have a chart of Europe and a chart of Russia, then we can compare these two charts on their overlap, namely the European part of Russia.) To be more precise, suppose that (U_, \varphi_) and (U_, \varphi_) are two charts for a manifold ''M'' such that U_ \cap U_ is
non-empty In mathematics, the empty set is the unique set having no elements; its size or cardinality (count of elements in a set) is zero. Some axiomatic set theories ensure that the empty set exists by including an axiom of empty set, while in other t ...
. The transition map \tau_: \varphi_(U_ \cap U_) \to \varphi_(U_ \cap U_) is the map defined by \tau_ = \varphi_ \circ \varphi_^. Note that since \varphi_ and \varphi_ are both homeomorphisms, the transition map \tau_ is also a homeomorphism.


More structure

One often desires more structure on a manifold than simply the topological structure. For example, if one would like an unambiguous notion of differentiation of functions on a manifold, then it is necessary to construct an atlas whose transition functions are
differentiable In mathematics, a differentiable function of one real variable is a function whose derivative exists at each point in its domain. In other words, the graph of a differentiable function has a non-vertical tangent line at each interior point in its ...
. Such a manifold is called
differentiable In mathematics, a differentiable function of one real variable is a function whose derivative exists at each point in its domain. In other words, the graph of a differentiable function has a non-vertical tangent line at each interior point in its ...
. Given a differentiable manifold, one can unambiguously define the notion of
tangent vectors In mathematics, a tangent vector is a vector that is tangent to a curve or surface at a given point. Tangent vectors are described in the differential geometry of curves in the context of curves in R''n''. More generally, tangent vectors are eleme ...
and then
directional derivative In mathematics, the directional derivative of a multivariable differentiable (scalar) function along a given vector v at a given point x intuitively represents the instantaneous rate of change of the function, moving through x with a velocity ...
s. If each transition function is a
smooth map 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 ...
, then the atlas is called a
smooth atlas 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 A smooth structure on a manifold M is ...
, and the manifold itself is called
smooth Smooth may refer to: Mathematics * Smooth function, a function that is infinitely differentiable; used in calculus and topology * Smooth manifold, a differentiable manifold for which all the transition maps are smooth functions * Smooth algebrai ...
. Alternatively, one could require that the transition maps have only ''k'' continuous derivatives in which case the atlas is said to be C^k . Very generally, if each transition function belongs to a
pseudogroup In mathematics, a pseudogroup is a set of diffeomorphisms between open sets of a space, satisfying group-like and sheaf-like properties. It is a generalisation of the concept of a group, originating however from the geometric approach of Sophus Lie ...
\mathcal G of homeomorphisms of Euclidean space, then the atlas is called a \mathcal G-atlas. If the transition maps between charts of an atlas preserve a
local trivialization In mathematics, and particularly topology, a fiber bundle (or, in Commonwealth English: fibre bundle) is a space that is a product space, but may have a different topological structure. Specifically, the similarity between a space E and a ...
, then the atlas defines the structure of a fibre bundle.


See also

*
Smooth atlas 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 A smooth structure on a manifold M is ...
*
Smooth frame In mathematics, a moving frame is a flexible generalization of the notion of an ordered basis of a vector space often used to study the extrinsic differential geometry of smooth manifolds embedded in a homogeneous space. Introduction In lay t ...


References

* * * * *, Chapter 5 "Local coordinate description of fibre bundles".


External links


Atlas
by Rowland, Todd {{Manifolds Manifolds