Atlas (topology)

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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.

ChartsTemplate:Anchor

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>Template:Cite book</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>Template:Cite book</ref><ref>Template:Cite book</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:Template:Clarify

• 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,Template:Clarify
• <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">Template:Cite book</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

Template:Annotated image 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

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• Template:Citation, Chapter 5 "Local coordinate description of fibre bundles".

External links

• Atlas by Rowland, Todd