Atlas (topology)

Set of charts that describes a manifold

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

[edit]

"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 φ {\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  ( U , φ ) {\displaystyle (U,\varphi )} .[1]

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

## Formal definition of atlas

[edit]

An atlas for a topological space M {\displaystyle M} is an indexed family { ( U α , φ α ) : α ∈ I } {\displaystyle \\{(U_{\alpha },\varphi _{\alpha }):\alpha \in I\\}} of charts on  M {\displaystyle M} which covers M {\displaystyle M} (that is,  ⋃ α ∈ I U α = M {\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 {\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  ( U i , φ i ) i ∈ I {\displaystyle \left(U_{i},\varphi _{i}\right)_{i\in I}} on an  n {\displaystyle n} -dimensional manifold  M {\displaystyle M} is called an adequate atlas if the following conditions hold:[clarification needed]

* The image of each chart is either  R n {\displaystyle \mathbb {R} ^{n}} or  R + n {\displaystyle \mathbb {R} _{+}^{n}} , where  R + n {\displaystyle \mathbb {R} _{+}^{n}} is the closed half-space,[clarification needed]
* ( U i ) i ∈ I {\displaystyle \left(U_{i}\right)_{i\in I}} is a locally finite open cover of  M {\displaystyle M} , and
* M = ⋃ i ∈ I φ i − 1 ( B 1 ) {\textstyle M=\bigcup _{i\in I}\varphi _{i}^{-1}\left(B_{1}\right)} , where  B 1 {\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  V = ( V j ) j ∈ J {\displaystyle {\mathcal {V}}=\left(V_{j}\right)_{j\in J}} is an open covering of the second-countable manifold  M {\displaystyle M} , then there is an adequate atlas  ( U i , φ i ) i ∈ I {\displaystyle \left(U_{i},\varphi _{i}\right)_{i\in I}} on  M {\displaystyle M} , such that  ( U i ) i ∈ I {\displaystyle \left(U_{i}\right)_{i\in I}} is a refinement of  V {\displaystyle {\mathcal {V}}} .[4]

## Transition maps

[edit]

M {\displaystyle M}

U α {\displaystyle U_{\alpha }}

U β {\displaystyle U_{\beta }}

φ α {\displaystyle \varphi _{\alpha }}

φ β {\displaystyle \varphi _{\beta }}

τ α , β {\displaystyle \tau _{\alpha ,\beta }}

τ β , α {\displaystyle \tau _{\beta ,\alpha }}

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

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

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

## More structure

[edit]

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 {\displaystyle C^{k}} .

Very generally, if each transition function belongs to a pseudogroup G {\displaystyle {\mathcal {G}}} of homeomorphisms of Euclidean space, then the atlas is called a  G {\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.

## See also

[edit]

* Smooth atlas
* Smooth frame

## References

[edit]

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

* 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

[edit]

* Atlas by Rowland, Todd

* v
* t
* e

Manifolds (Glossary, List, Category)

Basic concepts|

* Topological manifold
* Atlas
* Differentiable/Smooth manifold
* Differential structure
* Smooth atlas
* Submanifold
* Riemannian manifold
* Smooth map
* Submersion
* Pushforward
* Tangent space
* Differential form
* Vector field

Main results (list)|

* Atiyah–Singer index
* Darboux's
* De Rham's
* Frobenius
* Generalized Stokes
* Hopf–Rinow
* Noether's
* Sard's
* Whitney embedding

Maps|

* Curve
* Diffeomorphism
* Local
* Geodesic
* Exponential map
* in Lie theory
* Foliation
* Immersion
* Integral curve
* Lie derivative
* Section
* Submersion

Types of
manifolds|

* Closed
* Collapsing
* Complete
* (Almost) Complex
* (Almost) Contact
* Fibered
* Finsler
* (Almost) Flat
* G-structure
* Hadamard
* Hermitian
* Hyperbolic
* Kähler
* Kenmotsu
* Lie group
* Lie algebra
* Manifold with boundary
* Nilmanifold
* Oriented
* Parallelizable
* Poisson
* Prime
* Quaternionic
* Hypercomplex
* (Pseudo-, Sub-) Riemannian
* Rizza
* Stein
* (Almost) Symplectic
* Tame

Tensors|

| Vectors|

* Distribution
* Lie bracket
* Pushforward
* Tangent space
* bundle
* Torsion
* Vector field
* Vector flow

|

Covectors|

* Closed/Exact
* Covariant derivative
* Cotangent space
* bundle
* De Rham cohomology
* Differential form
* Vector-valued
* Exterior derivative
* Interior product
* Pullback
* Ricci curvature
* flow
* Riemann curvature tensor
* Tensor field
* density
* Volume form
* Wedge product

Bundles|

* Adjoint
* Affine
* Associated
* Cotangent
* Dual
* Fiber
* (Co-) Fibration
* Jet
* Lie algebra
* (Stable) Normal
* Principal
* Spinor
* Subbundle
* Tangent
* Tensor
* Vector

Connections|

* Affine
* Cartan
* Ehresmann
* Form
* Generalized
* Koszul
* Levi-Civita
* Principal
* Vector
* Parallel transport

Related|

* Classification of manifolds
* Gauge theory
* History
* Morse theory
* Moving frame
* Singularity theory

Generalizations|

* Banach manifold
* Diffeology
* Diffiety
* Fréchet manifold
* K-theory
* Orbifold
* Secondary calculus
* over commutative algebras
* Sheaf
* Stratifold
* Supermanifold
* Stratified space

*[v]: View this template
*[t]: Discuss this template
*[e]: Edit this template