Atlas (topology)

For other uses, see Fiber bundle and Atlas (disambiguation).

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

"Coordinate patch" redirects here; not to be confused with Surface patch.

"Local coordinate system" redirects here; not to be confused with Local geodetic coordinate system.

See also: Topological manifold § Coordinate 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, ${\textstyle \bigcup _{\alpha \in I}U_{\alpha }=M}$). If for some fixed n, the image of each chart is an open subset of n-dimensional Euclidean space, then $M$ 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 $\mathbb {R} ^{n}$ or $\mathbb {R} _{+}^{n}$, where $\mathbb {R} _{+}^{n}$ is the closed half-space,
• $\left(U_{i}\right)_{i\in I}$ is a locally finite open cover of $M$, and
• ${\textstyle M=\bigcup _{i\in I}\varphi _{i}^{-1}\left(B_{1}\right)}$, 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

$M$

$U_{\alpha }$

$U_{\beta }$

$\varphi _{\alpha }$

$\varphi _{\beta }$

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

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

$\mathbf {R} ^{n}$

$\mathbf {R} ^{n}$

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 $(U_{\alpha },\varphi _{\alpha })$ 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.

See also

• Smooth atlas
• Smooth frame

References

• Dieudonné, Jean (1972). "XVI. Differential manifolds". Treatise on Analysis. Pure and Applied Mathematics. Vol. III. Translated by Ian G. Macdonald. Academic Press. MR 0350769.
• Lee, John M. (2006). Introduction to Smooth Manifolds. Springer-Verlag. ISBN 978-0-387-95448-6.
• Loomis, Lynn; Sternberg, Shlomo (2014). "Differentiable manifolds". Advanced Calculus (Revised ed.). World Scientific. pp. 364–372. ISBN 978-981-4583-93-0. MR 3222280.
• Sepanski, Mark R. (2007). Compact Lie Groups. Springer-Verlag. ISBN 978-0-387-30263-8.
• Husemoller, D (1994), Fibre bundles, Springer, Chapter 5 "Local coordinate description of fibre bundles".

External links

• Atlas by Rowland, Todd