 Atlas (topology)

 For other uses of "atlas", see Atlas (disambiguation).
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.
Contents
Charts
The definition of an atlas depends on the notion of a chart. A chart for a topological space M is a homeomorphism ϕ from an open subset U of M to an open subset of Euclidean space. The chart is traditionally recorded as the ordered pair (U,φ).
Formal defintion of atlas
An atlas for a topological space M is a collection {(U_{α},φ_{α})} of charts on M such that . If the range of each chart is the ndimensional Euclidean space, then M is said to be an ndimensional manifold.
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 welldefined 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 (U_{α},φ_{α}) and (U_{β},φ_{β}) are two charts for a manifold M such that is nonempty. The transition map τ_{α,β} is the map defined on the intersection by
Note that since φ_{α} and φ_{β} are both homeomorphisms, the transition map τ_{α,β} 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. Then 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. 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 of homeomorphisms of Euclidean space, then the atlas is called a atlas.
References
External links
 Atlas by Rowland, Todd
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