Atlas (topology)

Summary

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 edit

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 edit

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 second-countable manifold admits an adequate atlas.[3] Moreover, if   is an open covering of the second-countable manifold  , then there is an adequate atlas   on  , such that   is a refinement of  .[3]


Transition maps edit

 
 
 
 
 
 
 
 
 
 
 
Two charts on a manifold, and their respective transition map

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 edit

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  .

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 local trivialization, then the atlas defines the structure of a fibre bundle.

See also edit

References edit

  1. ^ Jost, Jürgen (11 November 2013). Riemannian Geometry and Geometric Analysis. Springer Science & Business Media. ISBN 9783662223857. Retrieved 16 April 2018 – via Google Books.
  2. ^ Giaquinta, Mariano; Hildebrandt, Stefan (9 March 2013). Calculus of Variations II. Springer Science & Business Media. ISBN 9783662062012. Retrieved 16 April 2018 – via Google Books.
  3. ^ a b Kosinski, Antoni (2007). Differential manifolds. Mineola, N.Y: Dover Publications. ISBN 978-0-486-46244-8. OCLC 853621933.
  • Dieudonné, Jean (1972). "XVI. Differential manifolds". Treatise on Analysis. Pure and Applied Mathematics. Vol. III. Translated by Ian G. Macdonald. Academic Press. MR 0350769.
  • Lee, John M. (2006). Introduction to Smooth Manifolds. Springer-Verlag. ISBN 978-0-387-95448-6.
  • Loomis, Lynn; Sternberg, Shlomo (2014). "Differentiable manifolds". Advanced Calculus (Revised ed.). World Scientific. pp. 364–372. ISBN 978-981-4583-93-0. 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 edit

  • Atlas by Rowland, Todd