Atlas (topology)
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 (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
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
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
If each transition function is a
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
See also
- Smooth atlas
- Smooth frame
References
- ISBN 9783662223857. Retrieved 16 April 2018 – via Google Books.
- ISBN 9783662062012. Retrieved 16 April 2018 – via Google Books.
- ^ OCLC 853621933.
- MR 0350769.
- Lee, John M. (2006). Introduction to Smooth Manifolds. Springer-Verlag. ISBN 978-0-387-95448-6.
- MR 3222280.
- Sepanski, Mark R. (2007). Compact Lie Groups. Springer-Verlag. ISBN 978-0-387-30263-8.
- Husemoller, D (1994), Fibre bundles, Springer, Chapter 5 "Local coordinate description of fibre bundles".
External links
- Atlas by Rowland, Todd