Transition map

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.

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 ${\displaystyle \varphi }$ from an open subset U of M to an open subset of a Euclidean space. The chart is traditionally recorded as the ordered pair ${\displaystyle (U,\varphi )}$.

An atlas for a topological space M is a collection ${\displaystyle \{(U_{\alpha },\varphi _{\alpha })\mid \alpha \in A\},}$ indexed by a set A, of charts on M such that ${\displaystyle \textstyle \bigcup _{\alpha \in A}U_{\alpha }=M}$. If the codomain of each chart is the n-dimensional Euclidean space, then M is said to be an n-dimensional manifold.

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