Editing Atlas (topology)

{{short description|Set of charts that describes a manifold}}
{{other uses|Fiber bundle|Atlas (disambiguation)}}

In [[mathematics]], particularly [[topology]], an '''atlas''' is a concept used to describe a [[manifold]]. An atlas consists of individual ''charts'' that, roughly speaking, describe individual regions of the manifold. In general, the notion of atlas underlies the formal definition of a [[manifold]] and related structures such as [[vector bundle]]s and other [[fiber bundle]]s.

==Charts{{anchor|Maps}}==
{{redirect-distinguish|Coordinate patch|Surface patch}}
{{redirect-distinguish|Local coordinate system|Local geodetic coordinate system}}
{{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'' 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>.<ref>{{cite book |last1=Jänich |first1=Klaus |title=Vektoranalysis |date=2005 |publisher=Springer |isbn=3-540-23741-0 |page=1 |edition=5 |language=German}}</ref>

When a coordinate system is chosen in the Euclidean space, this defines coordinates on <math>U</math>: the coordinates of a point <math>P</math> of <math>U</math> are defined as the coordinates of <math>\varphi(P).</math> The pair formed by a chart and such a coordinate system is called a '''local coordinate system''', '''coordinate chart''', '''coordinate patch''', '''coordinate map''', or '''local frame'''.

==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>M</math> (that is, <math display="inline">\bigcup_{\alpha\in I} U_{\alpha} = M</math>). If for some fixed ''n'', the [[image (mathematics)|image]] of each chart is an open subset of ''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 book|url=https://books.google.com/books?id=VRz2CAAAQBAJ&pg=PA1| title=Riemannian Geometry and Geometric Analysis|first=Jürgen|last=Jost|date=11 November 2013| publisher=Springer Science & Business Media|isbn=9783662223857|access-date=16 April 2018|via=Google Books}}</ref><ref>{{cite book| url=https://books.google.com/books?id=_ZT_CAAAQBAJ&pg=PA418|title=Calculus of Variations II|first1=Mariano|last1=Giaquinta| first2=Stefan|last2=Hildebrandt|date=9 March 2013|publisher=Springer Science & Business Media|isbn=9783662062012|access-date=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 following conditions hold:{{clarify|reason=why not restricting the charts to subsets whose images are unit balls, that is, defining adequate as "locally finite cover by open charts whose images are unit open balls"|date=May 2024}}

* The [[image (mathematics)|image]] of each chart is either <math>\R^n</math> or <math>\R_+^n</math>, where <math>\R_+^n</math> is the [[closed half-space]],{{clarify|reason=the image of a chart must be open|date=May 2024}}
* <math>\left( U_i \right)_{i \in I}</math> is a [[Locally finite collection|locally finite]] open cover of <math>M</math>, and
* <math display="inline">M = \bigcup_{i \in I} \varphi_i^{-1}\left( B_1 \right)</math>, where <math>B_1</math> is the open ball of radius 1 centered at the origin.

Every [[second-countable]] manifold admits an adequate atlas.<ref name="Kosinski 2007">{{cite book | last=Kosinski | first=Antoni | title=Differential manifolds | publisher=Dover Publications | location=Mineola, N.Y | year=2007 | isbn=978-0-486-46244-8 | oclc=853621933 }}</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 a cover|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 display="block">\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|mr=0350769|last=Dieudonné|first=Jean|author-link=Jean Dieudonné| title=[[Treatise on Analysis]] | chapter=XVI. Differential manifolds| volume= III|translator= [[Ian G. Macdonald]]|series=Pure and Applied Mathematics|publisher=[[Academic Press]] | year=1972}}
*{{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|first1=Lynn|last1=Loomis|author1-link=Lynn Loomis|first2=Shlomo|last2=Sternberg|author2-link=Shlomo Sternberg | title=Advanced Calculus|edition=Revised|year=2014|publisher=World Scientific | isbn=978-981-4583-93-0 | mr=3222280 | chapter=Differentiable manifolds|pages=364–372}}
*{{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

{{Manifolds}}

[[Category:Manifolds]]