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

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>.

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.

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:

  • 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,
  • <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. 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>.