Transition map

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

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

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

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

Transition maps

M {\displaystyle M} {\displaystyle M}
U α {\displaystyle U_{\alpha }} {\displaystyle U_{\alpha }}
U β {\displaystyle U_{\beta }} {\displaystyle U_{\beta }}
φ α {\displaystyle \varphi _{\alpha }} {\displaystyle \varphi _{\alpha }}
φ β {\displaystyle \varphi _{\beta }} {\displaystyle \varphi _{\beta }}
τ α , β {\displaystyle \tau _{\alpha ,\beta }} {\displaystyle \tau _{\alpha ,\beta }}
τ β , α {\displaystyle \tau _{\beta ,\alpha }} {\displaystyle \tau _{\beta ,\alpha }}
R n {\displaystyle \mathbf {R} ^{n}} {\displaystyle \mathbf {R} ^{n}}
R n {\displaystyle \mathbf {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 })} {\displaystyle (U_{\alpha },\varphi _{\alpha })} and ( U β , φ β ) {\displaystyle (U_{\beta },\varphi _{\beta })} {\displaystyle (U_{\beta },\varphi _{\beta })} are two charts for a manifold M such that U α ∩ U β {\displaystyle U_{\alpha }\cap U_{\beta }} {\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 })} {\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}.} {\displaystyle \tau _{\alpha ,\beta }=\varphi _{\beta }\circ \varphi _{\alpha }^{-1}.}

Note that since φ α {\displaystyle \varphi _{\alpha }} {\displaystyle \varphi _{\alpha }} and φ β {\displaystyle \varphi _{\beta }} {\displaystyle \varphi _{\beta }} are both homeomorphisms, the transition map τ α , β {\displaystyle \tau _{\alpha ,\beta }} {\displaystyle \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 {\displaystyle C^{k}} {\displaystyle C^{k}}.

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

References

  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. Kosinski, Antoni (2007). Differential manifolds. Mineola, N.Y: Dover Publications. ISBN 978-0-486-46244-8. OCLC 853621933.