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

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 $\phi (P).$ 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 $\{(U_{\alpha },{\phi }_{\alpha }):\alpha \in I\}$ 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 ${(U_i,{\phi }_i)}_{i\in I}$ on an $n$-dimensional manifold $M$ is called an adequate atlas if the following conditions hold:Template:Clarify

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

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

Transition maps

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

Note that since ${\phi }_{\alpha }$ and ${\phi }_{\beta }$ 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 ${𝒢}$ of homeomorphisms of Euclidean space, then the atlas is called a ${𝒢}$-atlas. If the transition maps between charts of an atlas preserve a local trivialization, then the atlas defines the structure of a fibre bundle.

See also

• Smooth atlas
• Smooth frame

References

• , Chapter 5 "Local coordinate description of fibre bundles".

External links

• Atlas by Rowland, Todd