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| : ''For other uses of "atlas", see [[Atlas (disambiguation)]].''
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| In [[mathematics]], particularly [[topology]], one describes
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| a [[manifold]] using an '''atlas'''. An atlas consists of individual
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| ''charts'' that, roughly speaking, describe individual regions
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| of the manifold. If the manifold is the surface of the Earth,
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| then an atlas has its more common meaning. In general,
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| the notion of atlas underlies the formal definition of a [[manifold]].
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| ==Charts==
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| The definition of an atlas depends on the notion of a ''chart''.
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| 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 [[Euclidean space]]. The chart is traditionally recorded as the ordered pair <math> (U, \varphi)</math>.
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| ==Formal definition of atlas==
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| An '''atlas''' for a [[topological space]] ''M'' is a collection <math> \{(U_{\alpha}, \varphi_{\alpha})\}</math> of charts on ''M'' such that
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| <math> \bigcup U_{\alpha} = M</math>. If the codomain of each chart is the ''n''-dimensional [[Euclidean space]] and the atlas is connected, then ''M'' is said to be an ''n''-dimensional [[manifold]].
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| ==Transition maps==
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| {{ Annotated image | caption=Two charts on a manifold
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| | image=Two coordinate charts on a manifold.svg
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| | image-width = 250
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| | annotations =
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| {{Annotation|45|70|<math>M</math>}}
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| {{Annotation|67|54|<math>U_\alpha</math>}}
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| {{Annotation|187|66|<math>U_\beta</math>}}
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| {{Annotation|42|100|<math>\varphi_\alpha</math>}}
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| {{Annotation|183|117|<math>\varphi_\beta</math>}}
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| {{Annotation|87|109|<math>\tau_{\alpha,\beta}</math>}}
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| {{Annotation|90|145|<math>\tau_{\beta,\alpha}</math>}}
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| {{Annotation|55|183|<math>\mathbf R^n</math>}}
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| {{Annotation|145|183|<math>\mathbf R^n</math>}}
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| }}
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| A transition map provides a way of comparing two charts of an atlas.
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| To make this comparison, we consider the composition of one chart
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| with the inverse of the other. This composition is not well-defined
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| unless we restrict both charts to the intersection of their domains
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| 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.)
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| 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.
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| 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
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| : <math>\tau_{\alpha,\beta} = \varphi_{\beta} \circ \varphi_{\alpha}^{-1}.</math> | |
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| 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.
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| ==More structure==
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| 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]]. 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.
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| If each transition function
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| is a [[smooth map]], then the atlas is called a
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| [[smooth structure|smooth atlas]], and the manifold itself is called [[Differentiable manifold#Definition|smooth]].
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| Alternatively, one could require that the transition maps
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| have only ''k'' continuous derivatives in which case the atlas is
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| said to be <math> C^k </math>.
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| Very generally, if each transition function
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| belongs to a [[pseudo-group]] <math> {\mathcal G} </math>
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| of [[homeomorphism]]s of [[Euclidean space]],
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| then the atlas is called a <math> {\mathcal G}</math>-atlas.
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| ==References==
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| {{reflist}}
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| {{refbegin}}
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| *{{cite book | first = John M. | last = Lee | year = 2006 | title = Introduction to Smooth Manifolds | publisher = Springer-Verlag | isbn = 978-0-387-95448-6}}
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| *{{cite book | first = Mark R. | last = Sepanski | year = 2007 | title = Compact Lie Groups | publisher = Springer-Verlag | isbn = 978-0-387-30263-8}}
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| {{refend}}
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| ==External links==
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| *[http://mathworld.wolfram.com/Atlas.html Atlas] by Rowland, Todd
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| [[Category:Differential topology]]
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