Atlas (topology)#Transition maps

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|Local coordinate system|Local geodetic coordinate system}}

The definition of an atlas depends on the notion of a chart. A chart for a topological space M is a homeomorphism $\varphi$ from an open subset U of M to an open subset of a Euclidean space. The chart is traditionally recorded as the ordered pair $(U, \varphi)$ .{{cite book |last1=Jänich |first1=Klaus |title=Vektoranalysis |date=2005 |publisher=Springer |isbn=3-540-23741-0 |page=1 |edition=5 |language=German}}

When a coordinate system is chosen in the Euclidean space, this defines coordinates on $U$ : the coordinates of a point $P$ of $U$ are defined as the coordinates of $\varphi(P).$ 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 $M$ is an indexed family $\{(U_{\alpha}, \varphi_{\alpha}) : \alpha \in I\}$ of charts on $M$ which covers $M$ (that is, $\bigcup_{\alpha\in I} U_{\alpha} = M$ ). If for some fixed n, the image of each chart is an open subset of n-dimensional Euclidean space, then $M$ is said to be an n-dimensional manifold.

The plural of atlas is atlases, although some authors use atlantes.{{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}}{{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}}

An atlas $\left( U_i, \varphi_i \right)_{i \in I}$ on an $n$ -dimensional manifold $M$ 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 $\R^n$ or $\R_+^n$ , where $\R_+^n$ is the closed half-space,{{clarify|reason=the image of a chart must be open|date=May 2024}}
$\left( U_i \right)_{i \in I}$ is a locally finite open cover of $M$ , and
$M = \bigcup_{i \in I} \varphi_i^{-1}\left( B_1 \right)$ , where $B_1$ is the open ball of radius 1 centered at the origin.

Every second-countable manifold admits an adequate atlas.{{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 }} Moreover, if $\mathcal{V} = \left( V_j \right)_{j \in J}$ is an open covering of the second-countable manifold $M$ , then there is an adequate atlas $\left( U_i, \varphi_i \right)_{i \in I}$ on $M$ , such that $\left( U_i\right)_{i \in I}$ is a refinement of $\mathcal{V}$ .

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|87|109| $\tau_{\alpha,\beta}$ }}

{{Annotation|90|145| $\tau_{\beta,\alpha}$ }}

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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 $(U_{\alpha}, \varphi_{\alpha})$ and $(U_{\beta}, \varphi_{\beta})$ are two charts for a manifold M such that $U_{\alpha} \cap U_{\beta}$ is non-empty.

The transition map $\tau_{\alpha,\beta}: \varphi_{\alpha}(U_{\alpha} \cap U_{\beta}) \to \varphi_{\beta}(U_{\alpha} \cap U_{\beta})$ is the map defined by

$\tau_{\alpha,\beta} = \varphi_{\beta} \circ \varphi_{\alpha}^{-1}.$

Note that since $\varphi_{\alpha}$ and $\varphi_{\beta}$ are both homeomorphisms, the transition map $\tau_{\alpha, \beta}$ 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 $C^k$ .

Very generally, if each transition function belongs to a pseudogroup $\mathcal G$ of homeomorphisms of Euclidean space, then the atlas is called a $\mathcal G$ -atlas. If the transition maps between charts of an atlas preserve a local trivialization, then the atlas defines the structure of a fibre bundle.

References

{{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".

External links

[http://mathworld.wolfram.com/Atlas.html Atlas] by Rowland, Todd

Category:Manifolds