# Atlas (topology)

# Atlas (topology)

In mathematics, particularly topology, one describes a manifold using an **atlas**. An atlas consists of individual *charts* that, roughly speaking, describe individual regions of the manifold. If the manifold is the surface of the Earth, then an atlas has its more common meaning. In general, the notion of atlas underlies the formal definition of a manifold and related structures such as vector bundles and other fibre bundles.

Charts

`The definition of an atlas depends on the notion of a`

*chart*. A**chart**for atopological space*M*(also called a**coordinate chart**,**coordinate patch**,**coordinate map**, or**local frame**) is ahomeomorphismfrom anopen subset*U*of*M*to an open subset of aEuclidean space. The chart is traditionally recorded as the ordered pair.Formal definition of atlas

`An`

**atlas**for atopological space*M*is a collectionindexed by a set*A*, of charts on*M*such that. If the codomain of each chart is the*n*-dimensionalEuclidean space, then*M*is said to be an*n*-dimensionalmanifold.The plural of atlas is *atlases*, although some authors use *atlantes*.^{[1]}^{[2]}

`An atlason an-dimensional manifoldis called an`

**adequate atlas**if the image of each chart is eitheror,is a locally finite open cover of, and, whereis the open ball of radius 1 centered at the origin andis the closed half space. Every second countable manifold admits an adequate atlas.^{[3]}Moreover, ifis an open covering of the second countable manifoldthen there is an adequate atlasonsuch thatis a refinement of.^{[3]}Transition maps

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 thatandare two charts for a manifold`

*M*such thatis non-empty. The**transition map**is the map defined by`Note that sinceandare both homeomorphisms, the transition mapis 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 asmooth map, then the atlas is called asmooth atlas, and the manifold itself is calledsmooth. 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 apseudogroupofhomeomorphismsofEuclidean space, then the atlas is called a-atlas. If the transition maps between charts of an atlas preserve alocal trivialization, then the atlas defines the structure of afibre bundle.`

See also

Smooth atlas

Smooth frame

## References

*Differential manifolds*. Mineola, N.Y: Dover Publications. ISBN 978-0-486-46244-8. OCLC 853621933.