View source for Atlas (topology)

{{other uses|Fiber bundle|Atlas (disambiguation)}}

In [[mathematics]], particularly [[topology]], 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]]s and other [[fibre bundle]]s.

==Charts{{anchor|Maps}}==
{{see also|Topological manifold#Coordinate_charts}}
The definition of an atlas depends on the notion of a ''chart''. A '''chart''' for a [[topological space]] ''M'' (also called a '''coordinate chart''', '''coordinate patch''', '''coordinate map''', or '''local frame''') is a [[homeomorphism]] <math>\varphi</math> from an [[open set|open subset]] ''U'' of ''M'' to an open subset of a [[Euclidean space]]. The chart is traditionally recorded as the ordered pair <math> (U, \varphi)</math>.

==Formal definition of atlas==
An '''atlas''' for a [[topological space]] <math>M</math> is an [[indexed family]] <math>\{(U_{\alpha}, \varphi_{\alpha}) : \alpha \in I\}</math> of charts on <math>M</math> which [[Cover (topology)|covers]] <math>X</math> (that is, <math>\textstyle \bigcup_{\alpha\in A} U_{\alpha} = M</math>). If the [[codomain]] of each chart is the ''n''-dimensional [[Euclidean space]], then <math>M</math> is said to be an ''n''-dimensional [[manifold]]. 

The plural of atlas is ''atlases'', although some authors use ''atlantes''.<ref>{{cite web|url=https://books.google.com/books?id=VRz2CAAAQBAJ&pg=PA1&lpg=PA1|title=Riemannian Geometry and Geometric Analysis|first=Jürgen|last=Jost|date=11 November 2013|publisher=Springer Science & Business Media|accessdate=16 April 2018|via=Google Books}}</ref><ref>{{cite web|url=https://books.google.com/books?id=_ZT_CAAAQBAJ&pg=PA418&lpg=PA418|title=Calculus of Variations II|first1=Mariano|last1=Giaquinta|first2=Stefan|last2=Hildebrandt|date=9 March 2013|publisher=Springer Science & Business Media|accessdate=16 April 2018|via=Google Books}}</ref>

An atlas <math>\left( U_i, \varphi_i \right)_{i \in I}</math> on an <math>n</math>-dimensional manifold <math>M</math> is called an '''adequate atlas''' if the [[Image (mathematics)|image]] of each chart is either <math>\mathbb{R}^n</math> or <math>\mathbb{R}^n_{+}</math>, <math>\left( U_i \right)_{i \in I}</math> is a [[Locally finite collection|locally finite]] open cover of <math>M</math>, and <math>M = \bigcup_{i \in I} \varphi^{-1}\left( B_1 \right)</math>, where <math>B_1</math> is the open ball of radius 1 centered at the origin and <math>\mathbb{R}^n_{+}</math> is the closed half space. Every [[second-countable]] manifold admits an adequate atlas.<ref name="Kosinski 2007">{{cite book | last=Kosinski | first=Antoni | title=Differential manifolds | publisher=Dover Publications | publication-place=Mineola, N.Y | year=2007 | isbn=978-0-486-46244-8 | oclc=853621933 | page=}}</ref> Moreover, if <math>\mathcal{V} = \left( V_j \right)_{j \in J}</math> is an open covering of the second-countable manifold <math>M</math> then there is an adequate atlas <math>\left( U_i, \varphi_i \right)_{i \in I}</math> on <math>M</math> such that <math>\left( U_i\right)_{i \in I}</math> is a refinement of <math>\mathcal{V}</math>.<ref name="Kosinski 2007"/>

==Transition maps==
{{ Annotated image | caption=Two charts on a manifold, and their respective '''transition map'''
| image=Two coordinate charts on a manifold.svg
| image-width = 250
| annotations =
{{Annotation|45|70|<math>M</math>}}
{{Annotation|67|54|<math>U_\alpha</math>}}
{{Annotation|187|66|<math>U_\beta</math>}}
{{Annotation|42|100|<math>\varphi_\alpha</math>}}
{{Annotation|183|117|<math>\varphi_\beta</math>}}
{{Annotation|87|109|<math>\tau_{\alpha,\beta}</math>}}
{{Annotation|90|145|<math>\tau_{\beta,\alpha}</math>}}
{{Annotation|55|183|<math>\mathbf R^n</math>}}
{{Annotation|145|183|<math>\mathbf R^n</math>}}
}}
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 function|inverse]] of the other. This composition is not well-defined unless we restrict both charts to the [[Intersection (set theory)|intersection]] of their [[Domain of a function|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 <math>(U_{\alpha}, \varphi_{\alpha})</math> and <math>(U_{\beta}, \varphi_{\beta})</math> are two charts for a manifold ''M'' such that <math>U_{\alpha} \cap U_{\beta}</math> is [[Empty set|non-empty]].
The '''transition map''' <math> \tau_{\alpha,\beta}: \varphi_{\alpha}(U_{\alpha} \cap U_{\beta}) \to \varphi_{\beta}(U_{\alpha} \cap U_{\beta})</math> is the map defined by

: <math>\tau_{\alpha,\beta} = \varphi_{\beta} \circ \varphi_{\alpha}^{-1}.</math>

Note that since <math>\varphi_{\alpha}</math> and <math>\varphi_{\beta}</math> are both homeomorphisms, the transition map <math> \tau_{\alpha, \beta}</math> 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 (mathematics)|differentiation]] of functions on a manifold, then it is necessary to construct an atlas whose transition functions are [[Differentiable function|differentiable]]. Such a manifold is called [[Differentiable manifold|differentiable]]. Given a differentiable manifold, one can unambiguously define the notion of [[tangent vectors]] and then [[directional derivative]]s.

If each transition function is a [[smooth map]], then the atlas is called a [[smooth structure|smooth atlas]], and the manifold itself is called [[Differentiable manifold#Definition|smooth]]. Alternatively, one could require that the transition maps have only ''k'' continuous derivatives in which case the atlas is said to be <math> C^k </math>.

Very generally, if each transition function belongs to a [[pseudogroup]] <math> {\mathcal G} </math> of homeomorphisms of Euclidean space, then the atlas is called a <math> {\mathcal G}</math>-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]]

==References==
{{reflist}}
{{refbegin}}
*{{cite book | first = John M. | last = Lee | year = 2006 | title = Introduction to Smooth Manifolds | publisher = Springer-Verlag | isbn = 978-0-387-95448-6}}
*{{cite book | first = Mark R. | last = Sepanski | year = 2007 | title = Compact Lie Groups | publisher = Springer-Verlag | isbn = 978-0-387-30263-8}}
*{{citation| last=Husemoller | first=D|title=Fibre bundles|publisher=Springer|year=1994}}, Chapter 5 "Local coordinate description of fibre bundles".
{{refend}}

==External links==
*[http://mathworld.wolfram.com/Atlas.html Atlas] by Rowland, Todd

[[Category:Manifolds]]