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.

The definition of an atlas depends on the notion of a chart. A chart for a topological space M is a homeomorphism ${\displaystyle \varphi }$ from an open subset U of M to an open subset of a Euclidean space. The chart is traditionally recorded as the ordered pair ${\displaystyle (U,\varphi )}$.^[1]

When a coordinate system is chosen in the Euclidean space, this defines coordinates on ${\displaystyle U}$: the coordinates of a point ${\displaystyle P}$ of ${\displaystyle U}$ are defined as the coordinates of ${\displaystyle \varphi (P).}$ 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.

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

The plural of atlas is atlases, although some authors use atlantes.^[2][3]

An atlas ${\displaystyle \left(U_{i},\varphi _{i}\right)_{i\in I}}$ on an ${\displaystyle n}$-dimensional manifold ${\displaystyle M}$ is called an adequate atlas if the following conditions hold:^{[clarification needed]}

• The image of each chart is either ${\displaystyle \mathbb {R} ^{n}}$ or ${\displaystyle \mathbb {R} _{+}^{n}}$, where ${\displaystyle \mathbb {R} _{+}^{n}}$ is the closed half-space,^{[clarification needed]}
• ${\displaystyle \left(U_{i}\right)_{i\in I}}$ is a locally finite open cover of ${\displaystyle M}$, and
• ${\textstyle M=\bigcup _{i\in I}\varphi _{i}^{-1}\left(B_{1}\right)}$, where ${\displaystyle B_{1}}$ is the open ball of radius 1 centered at the origin.

Every second-countable manifold admits an adequate atlas.^[4] Moreover, if ${\displaystyle {\mathcal {V}}=\left(V_{j}\right)_{j\in J}}$ is an open covering of the second-countable manifold ${\displaystyle M}$, then there is an adequate atlas ${\displaystyle \left(U_{i},\varphi _{i}\right)_{i\in I}}$ on ${\displaystyle M}$, such that ${\displaystyle \left(U_{i}\right)_{i\in I}}$ is a refinement of ${\displaystyle {\mathcal {V}}}$.^[4]

${\displaystyle M}$

${\displaystyle U_{\alpha }}$

${\displaystyle U_{\beta }}$

${\displaystyle \varphi _{\alpha }}$

${\displaystyle \varphi _{\beta }}$

${\displaystyle \tau _{\alpha ,\beta }}$

${\displaystyle \tau _{\beta ,\alpha }}$

${\displaystyle \mathbf {R} ^{n}}$

${\displaystyle \mathbf {R} ^{n}}$

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

Note that since ${\displaystyle \varphi _{\alpha }}$ and ${\displaystyle \varphi _{\beta }}$ are both homeomorphisms, the transition map ${\displaystyle \tau _{\alpha ,\beta }}$ is also a homeomorphism.

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 ${\displaystyle C^{k}}$.

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

• Smooth atlas
• Smooth frame

• Atlas by Rowland, Todd