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 is a homeomorphism $$\varphi$$ from an open subset U of M to an open subset of a Euclidean space. The chart is traditionally recorded as the ordered pair $$(U, \varphi)$$.

When a coordinate system is chosen in the Euclidean space, this defines coordinates on $$U$$: the coordinates of a point $$P$$ of $$U$$ are defined as the coordinates of $$\varphi(P).$$ The pair formed by a chart and such a coordinate system is called a local coordinate system, coordinate chart, coordinate patch, coordinate map, or local frame.

Formal definition of atlas
An atlas for a topological space $$M$$ is an indexed family $$\{(U_{\alpha}, \varphi_{\alpha}) : \alpha \in I\}$$ of charts on $$M$$ which covers $$M$$ (that is, $\bigcup_{\alpha\in I} U_{\alpha} = M$ ). If for some fixed n, the image of each chart is an open subset of n-dimensional Euclidean space, then $$M$$ is said to be an n-dimensional manifold.

The plural of atlas is atlases, although some authors use atlantes.

An atlas $$\left( U_i, \varphi_i \right)_{i \in I}$$ on an $$n$$-dimensional manifold $$M$$ is called an adequate atlas if the following conditions hold:


 * The image of each chart is either $$\R^n$$ or $$\R_+^n$$, where $$\R_+^n$$ is the closed half-space,
 * $$\left( U_i \right)_{i \in I}$$ is a locally finite open cover of $$M$$, and
 * $M = \bigcup_{i \in I} \varphi_i^{-1}\left( B_1 \right)$, where $$B_1$$ is the open ball of radius 1 centered at the origin.

Every second-countable manifold admits an adequate atlas. Moreover, if $$\mathcal{V} = \left( V_j \right)_{j \in J}$$ is an open covering of the second-countable manifold $$M$$, then there is an adequate atlas $$\left( U_i, \varphi_i \right)_{i \in I}$$ on $$M$$, such that $$\left( U_i\right)_{i \in I}$$ is a refinement of $$\mathcal{V}$$.

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 $$(U_{\alpha}, \varphi_{\alpha})$$ and $$(U_{\beta}, \varphi_{\beta})$$ are two charts for a manifold M such that $$U_{\alpha} \cap U_{\beta}$$ is non-empty. The transition map $$ \tau_{\alpha,\beta}: \varphi_{\alpha}(U_{\alpha} \cap U_{\beta}) \to \varphi_{\beta}(U_{\alpha} \cap U_{\beta})$$ is the map defined by $$\tau_{\alpha,\beta} = \varphi_{\beta} \circ \varphi_{\alpha}^{-1}.$$

Note that since $$\varphi_{\alpha}$$ and $$\varphi_{\beta}$$ are both homeomorphisms, the transition map $$ \tau_{\alpha, \beta}$$ 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. 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 $$ C^k $$.

Very generally, if each transition function belongs to a pseudogroup $$ \mathcal G$$ of homeomorphisms of Euclidean space, then the atlas is called a $$\mathcal G$$-atlas. If the transition maps between charts of an atlas preserve a local trivialization, then the atlas defines the structure of a fibre bundle.