Atlas (topology)
An atlas is a set of maps on a manifold. It serves to define a topological space, additional structures, such as a differentiable or a complex structure, so as to obtain a differentiable manifold or a complex manifold.
- 2.1 Differentiable structures
- 2.2 Complex structures
- 2.3 ( G, X ) structures
Definition
Map
Be a Hausdorff space, an open subset and an open subset of Euclidean space. A map is a homeomorphism on. To emphasize, to which basic amount is, you write the map as a 2- tuple.
It is possible to generalize this definition by choosing instead the room other rooms, such as the unitary vector space, a Banach space or a Hilbert space.
Atlas
More generally, an atlas on a lot of cards on which cover their domains of definition
If there exists such an atlas for a topological Hausdorff space, called this space manifold.
The homeomorphisms
Hot the cards transitions or switching maps of the atlas.
Additional Structures
With the help of an atlas, it is possible to define additional structures on a manifold. For example, you can try using the atlas to define a differentiable structure on the manifold. With this it is possible to explain differentiability of functions on the manifold. However, it may happen that certain cards are not compatible with each other, so that the choice of a differentiable structure may some maps from the Atlas must be removed. The property must remain, however obtained. An atlas that contains all cards that define the same differentiable structure, called Atlas maximum.
Differentiable structures
A differentiable atlas on a manifold is an atlas whose transition maps are diffeomorphisms.
A differentiable structure on a manifold is a maximal differentiable atlas.
A function is then called differentiable, if the picture is differentiable for a card. Because of the differentiability of the map transitions, this property does not depend on the choice of the map.
Complex structures
With the help of an atlas of maps with target area, one can try to define a complex structure on the manifold. Using this structure, it is possible to define holomorphic functions and meromorphic functions on a manifold and investigate.