Atlas (topology)

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 bundles and other fiber bundles.

==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 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 covers <math>M</math> (that is, <math display="inline">\bigcup_{\alpha\in I} U_{\alpha} = M</math>). If for some fixed ''n'', the 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|author-link=Jürgen 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 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 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 <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 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 <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 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 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 derivatives.

If each transition function is a smooth map, then the atlas is called a smooth atlas, and the manifold itself is called 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

Category:Manifolds