Atlas (mathematics)

An atlas is a set of maps on a manifold . It is used to define additional structures in a topological space, such as a differentiable or a complex structure, so that a differentiable manifold or a complex manifold is obtained.

definition

map

Let be a Hausdorff space , an open subset, and an open subset of Euclidean space . A map on is a homeomorphism . To emphasize which basic amount is involved, the card is also written as a pair . ${\ displaystyle S}$${\ displaystyle U \ subset S}$${\ displaystyle A \ subset \ mathbb {R} ^ {n}}$${\ displaystyle S}$ ${\ displaystyle \ phi \ colon U \ to A}$${\ displaystyle (U, \ phi)}$

It is possible to generalize this definition by choosing other spaces such as the unitary vector space , a Banach space or a Hilbert space instead of space . ${\ displaystyle \ mathbb {R} ^ {n}}$ ${\ displaystyle \ mathbb {C} ^ {n}}$

Atlas

In general, an atlas is based on a set of maps whose domains of definition cover: ${\ displaystyle S}$${\ displaystyle {\ mathcal {A}} = \ left \ {(U_ {i}, \ phi _ {i}) \ mid i \ in I \ right \}}$${\ displaystyle S}$${\ displaystyle S}$

${\ displaystyle S = \ bigcup _ {i \ in I} U_ {i}}$

If such an atlas exists for a topological Hausdorff space, this space is called a manifold.

The homeomorphisms

${\ displaystyle \ phi _ {i} \ circ \ phi _ {j} ^ {- 1} \ colon \ phi _ {j} (U_ {i} \ cap U_ {j}) \ to \ phi _ {i} (U_ {i} \ cap U_ {j})}$

are called the map transitions or map changes of the atlas.

With the help of an atlas it is possible to define additional structures on a manifold. For example, with the help of the atlas one can try to define a differentiable structure on the manifold. With this it is possible to explain the differentiability of functions on the manifold. However, it can happen that certain maps are incompatible with one another, so that when choosing a differentiable structure, certain maps may have to be removed from the atlas. However, the property must be retained. An atlas that contains all maps that define the same differentiable structure is called a maximal atlas. ${\ displaystyle \ textstyle S = \ bigcup _ {i \ in I} U_ {i}}$

Differentiable structures

A differentiable atlas on a manifold is an atlas whose map transitions are diffeomorphisms .

A differentiable structure on a manifold is a maximal differentiable atlas.

A function is then called differentiable in , if the mapping is differentiable for a map with . Because the map transitions can be differentiated, this property does not depend on the choice of map. ${\ displaystyle f \ colon M \ to \ mathbb {R}}$${\ displaystyle x}$${\ displaystyle (U_ {i}, \ phi _ {i})}$${\ displaystyle x \ in U_ {i}}$${\ displaystyle f \ circ \ phi _ {i} ^ {- 1}}$

Complex structures

With the help of an atlas of maps with a target area , one can define a complex structure on the manifold. With the help of this structure it is possible to define and investigate holomorphic functions and meromorphic functions on a manifold. ${\ displaystyle \ mathbb {C} ^ {n}}$

Individual evidence

1. ^ A b R. Abraham, JE Marsden, T. Ratiu: Manifolds, Tensor Analysis and Applications. 2nd Edition. Springer Verlag, Berlin 1993, ISBN 3-540-96790-7 .