# 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[edit]

The definition of an atlas depends on the notion of a *chart*. A **chart** for a topological space *M* (also called a **coordinate chart**, **coordinate patch**, **coordinate map**, or **local frame**) is a homeomorphism from an open subset *U* of *M* to an open subset of a Euclidean space. The chart is traditionally recorded as the ordered pair .

## Formal definition of atlas[edit]

An **atlas** for a topological space is an indexed family of charts on which covers (that is, ). If for some fixed *n*, the image of each chart is an open subset of *n*-dimensional Euclidean space, then is said to be an *n*-dimensional manifold.

The plural of atlas is *atlases*, although some authors use *atlantes*.^{[1]}^{[2]}

An atlas on an -dimensional manifold is called an **adequate atlas** if the following conditions hold:

- The image of each chart is either or , where is the closed half-space,
- is a locally finite open cover of , and
- , where is the open ball of radius 1 centered at the origin.

Every second-countable manifold admits an adequate atlas.^{[3]} Moreover, if is an open covering of the second-countable manifold , then there is an adequate atlas on , such that is a refinement of .^{[3]}

## Transition maps[edit]

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 and are two charts for a manifold *M* such that is non-empty.
The **transition map** is the map defined by

Note that since and are both homeomorphisms, the transition map is also a homeomorphism.

## More structure[edit]

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 .

Very generally, if each transition function belongs to a pseudogroup of homeomorphisms of Euclidean space, then the atlas is called a -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[edit]

## References[edit]

**^**Jost, Jürgen (11 November 2013).*Riemannian Geometry and Geometric Analysis*. Springer Science & Business Media. ISBN 9783662223857. Retrieved 16 April 2018 – via Google Books.**^**Giaquinta, Mariano; Hildebrandt, Stefan (9 March 2013).*Calculus of Variations II*. Springer Science & Business Media. ISBN 9783662062012. Retrieved 16 April 2018 – via Google Books.- ^
^{a}^{b}Kosinski, Antoni (2007).*Differential manifolds*. Mineola, N.Y: Dover Publications. ISBN 978-0-486-46244-8. OCLC 853621933.

- Dieudonné, Jean (1972). "XVI. Differential manifolds".
*Treatise on Analysis*. Pure and Applied Mathematics. Vol. III. Translated by Ian G. Macdonald. Academic Press. MR 0350769. - Lee, John M. (2006).
*Introduction to Smooth Manifolds*. Springer-Verlag. ISBN 978-0-387-95448-6. - Loomis, Lynn; Sternberg, Shlomo (2014). "Differentiable manifolds".
*Advanced Calculus*(Revised ed.). World Scientific. pp. 364–372. ISBN 978-981-4583-93-0. MR 3222280. - Sepanski, Mark R. (2007).
*Compact Lie Groups*. Springer-Verlag. ISBN 978-0-387-30263-8. - Husemoller, D (1994),
*Fibre bundles*, Springer, Chapter 5 "Local coordinate description of fibre bundles".

## External links[edit]

- Atlas by Rowland, Todd