Atlas (topology) explained

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

\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) .^[1]

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

M is an indexed family \{(U _{\alpha} , \varphi _{\alpha} ) : \alpha \in I\} of charts on M which covers M (that is, $\backslash bigcup\_\; U\_\; =\; M$). 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.^{[2] [3]}

An atlas

\left( U _{i,} \varphi _{i} \right) _{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} is a locally finite open cover of M , and

• $M\; =\; \backslash bigcup\_\; \backslash varphi\_i^\backslash left(B\_1\; \backslash right)$, where

B _{1} is the open ball of radius 1 centered at the origin.

Every second-countable manifold admits an adequate atlas.^[4] Moreover, if

l{V} = \left( V _{j} \right) _{j} is an open covering of the second-countable manifold M , then there is an adequate atlas \left( U _{i,} \varphi _{i} \right) _{i} on M , such that \left( U _{i\right)} _{i} is a refinement of l{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$$\backslash tau\_\; =\; \backslash varphi\_\; \backslash circ\; \backslash varphi\_^.$$

Note that since

\varphi _{\alpha} and \varphi _{\beta} are both homeomorphisms, the transition map \tau _{\alpha,} 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} .l G of homeomorphisms of Euclidean space, then the atlas is called a l 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

• Book: Dieudonné, Jean. 0350769. Jean Dieudonné. . XVI. Differential manifolds. III. Ian G. Macdonald. Pure and Applied Mathematics. . 1972.
• Book: Lee, John M. . 2006 . Introduction to Smooth Manifolds . Springer-Verlag . 978-0-387-95448-6.
• Book: Lynn. Loomis. Lynn Loomis. Shlomo. Sternberg. Shlomo Sternberg . Advanced Calculus. Revised. 2014. World Scientific . 978-981-4583-93-0 . 3222280 . Differentiable manifolds. 364–372.
• Book: Sepanski, Mark R. . 2007 . Compact Lie Groups . Springer-Verlag . 978-0-387-30263-8.
• , Chapter 5 "Local coordinate description of fibre bundles".

External links

• Atlas by Rowland, Todd

Notes and References

1. Book: Jänich . Klaus . Vektoranalysis . 2005 . Springer . 3-540-23741-0 . 1 . 5 . German.
2. Book: Jost, Jürgen. Riemannian Geometry and Geometric Analysis. 11 November 2013. Springer Science & Business Media. 9783662223857. 16 April 2018. Google Books.
3. Book: Calculus of Variations II. Mariano. Giaquinta. Stefan. Hildebrandt. 9 March 2013. Springer Science & Business Media. 9783662062012. 16 April 2018. Google Books.
4. Book: Kosinski, Antoni . Differential manifolds . Dover Publications . Mineola, N.Y . 2007 . 978-0-486-46244-8 . 853621933 .