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In mathematics, particularly topology, one describes 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 ...
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 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 ...
and related structures such as vector bundles 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 isomorph ...
\varphi from an open subset ''U'' of ''M'' to an open subset of a Euclidean space. 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 \ 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 ...
of each chart is the ''n''-dimensional Euclidean space, then M is said to be an ''n''-dimensional
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 ...
. 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 An image is a visual representation of something. It can be two-dimensional, three-dimensional, or somehow otherwise feed into the visual system to convey information. An image can be an artifact, such as a photograph or other two-dimensiona ...
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 Inverse or invert may refer to: Science and mathematics * Inverse (logic), a type of conditional sentence which is an immediate inference made from another conditional sentence * Additive inverse (negation), the inverse of a number that, when a ...
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. 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. Such a manifold is called differentiable. Given a differentiable manifold, one can unambiguously define the notion of tangent vectors 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, 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 algebraic ...
. 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 \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, then the atlas defines the structure of a fibre bundle.


See also

* Smooth atlas *
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 ...


References

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


External links


Atlas
by Rowland, Todd {{Manifolds Manifolds