GetWiki

{{short description|Set of charts that describes a manifold}}{{other uses|Fiber bundle|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{{anchor|Maps}}

{{redirect-distinguish|Coordinate patch|Surface patch}}{{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 (also called a coordinate chart, coordinate patch, coordinate map, or local frame) 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).

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_{alphain 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.BOOK,weblink Riemannian Geometry and Geometric Analysis, Jürgen, Jost, 11 November 2013, Springer Science & Business Media, 9783662223857, 16 April 2018, Google Books, BOOK,weblink Calculus of Variations II, Mariano, Giaquinta, Stefan, Hildebrandt, 9 March 2013, Springer Science & Business Media, 9783662062012, 16 April 2018, Google Books, 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.BOOK, Kosinski, Antoni, Differential manifolds, Dover Publications, Mineola, N.Y, 2007, 978-0-486-46244-8, 853621933, 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_iright)_{i in I} is a refinement of mathcal{V}.

Transition maps

{{Annotation|67|54|U_alpha}}{{Annotation|187|66|U_beta}}{{Annotation|42|100|varphi_alpha}}{{Annotation|183|117|varphi_beta}}{{Annotation|87|109|tau_{alpha,beta}}}{{Annotation|90|145|tau_{beta,alpha}}}{{Annotation|55|183|mathbf R^n}}{{Annotation|145|183|mathbf R^n}}}}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 bytau_{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

{{reflist}}

• BOOK, 0350769, Dieudonné, Jean, Jean Dieudonné, Treatise on Analysis, XVI. Differential manifolds, III, Ian G. Macdonald, Pure and Applied Mathematics, Academic Press, 1972,
• BOOK, John M., Lee, 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, Mark R., Sepanski, 2007, Compact Lie Groups, Springer-Verlag, 978-0-387-30263-8,
• {{citation| last=Husemoller | first=D|title=Fibre bundles|publisher=Springer|year=1994}}, Chapter 5 "Local coordinate description of fibre bundles".

External links

• Atlas by Rowland, Todd

{{Manifolds}}