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 \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. 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 of charts on M which covers M (that is, ). 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. An atlas on an n-dimensional manifold M is called an adequate atlas if the following conditions hold: Every second-countable manifold admits an adequate atlas. Moreover, if is an open covering of the second-countable manifold M, then there is an adequate atlas on M, such that is a refinement of \mathcal{V}.

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 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

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.