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.


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]

Two charts on a manifold, and their respective transition map

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]


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

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