Inmathematics, particularlytopology, anatlas is a concept used to describe amanifold. An atlas consists of individualcharts that, roughly speaking, describe individual regions of the manifold. In general, the notion of atlas underlies the formal definition of amanifold and related structures such asvector bundles and otherfiber bundles.

The definition of an atlas depends on the notion of achart. Achart for atopological spaceM is ahomeomorphism${\displaystyle \phi }$ from anopen subsetU ofM to an open subset of aEuclidean space. The chart is traditionally recorded as the ordered pair${\displaystyle (U,\phi )}$.^[1]

When a coordinate system is chosen in the Euclidean space, this defines coordinates on${\displaystyle U}$: the coordinates of a point${\displaystyle P}$ of${\displaystyle U}$ are defined as the coordinates of${\displaystyle \phi (P).}$ The pair formed by a chart and such a coordinate system is called alocal coordinate system,coordinate chart,coordinate patch,coordinate map, orlocal frame.

Anatlas for atopological space${\displaystyle M}$ is anindexed family${\displaystyle \{(U_{\alpha },{\phi }_{\alpha }):\alpha \in I\}}$ of charts on${\displaystyle M}$ whichcovers${\displaystyle M}$ (that is,$\bigcup _{\alpha \in I}{U}_{\alpha }=M$). If for some fixedn, theimage of each chart is an open subset ofn-dimensionalEuclidean space, then${\displaystyle M}$ is said to be ann-dimensionalmanifold.

The plural of atlas isatlases, although some authors useatlantes.^[2][3]

An atlas${\displaystyle {\left(U_i,{\phi }_i\right)}_{i\in I}}$ on an${\displaystyle n}$-dimensional manifold${\displaystyle M}$ is called anadequate atlas if the following conditions hold:^{[clarification needed]}

• Theimage of each chart is either${\displaystyle {\mathbb{R}}^n}$ or${\displaystyle {\mathbb{R}}_{+}^n}$, where${\displaystyle {\mathbb{R}}_{+}^n}$ is theclosed half-space,^{[clarification needed]}
• ${\displaystyle {\left(U_i\right)}_{i\in I}}$ is alocally finite open cover of${\displaystyle M}$, and
• $M=\bigcup _{i\in I}{\phi }_i^{-1}\left(B_1\right)$, where${\displaystyle B_1}$ is the open ball of radius 1 centered at the origin.

Everysecond-countable manifold admits an adequate atlas.^[4] Moreover, if${\displaystyle \mathcal{V}={\left(V_j\right)}_{j\in J}}$ is an open covering of the second-countable manifold${\displaystyle M}$, then there is an adequate atlas${\displaystyle {\left(U_i,{\phi }_i\right)}_{i\in I}}$ on${\displaystyle M}$, such that${\displaystyle {\left(U_i\right)}_{i\in I}}$ is arefinement of${\displaystyle \mathcal{V}}$.^[4]

${\displaystyle M}$

${\displaystyle U_{\alpha }}$

${\displaystyle U_{\beta }}$

${\displaystyle {\phi }_{\alpha }}$

${\displaystyle {\phi }_{\beta }}$

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

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

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

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

Two charts on a manifold, and their respectivetransition 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 theinverse of the other. This composition is not well-defined unless we restrict both charts to theintersection of theirdomains 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${\displaystyle (U_{\alpha },{\phi }_{\alpha })}$ and${\displaystyle (U_{\beta },{\phi }_{\beta })}$ are two charts for a manifoldM such that${\displaystyle U_{\alpha }\cap U_{\beta }}$ isnon-empty.Thetransition map${\displaystyle {\tau }_{\alpha ,\beta }:{\phi }_{\alpha }(U_{\alpha }\cap U_{\beta })\to {\phi }_{\beta }(U_{\alpha }\cap U_{\beta })}$ is the map defined by$${\displaystyle {\tau }_{\alpha ,\beta }={\phi }_{\beta }\circ {\phi }_{\alpha }^{-1}.}$$

Note that since${\displaystyle {\phi }_{\alpha }}$ and${\displaystyle {\phi }_{\beta }}$ are both homeomorphisms, the transition map${\displaystyle {\tau }_{\alpha ,\beta }}$ is also a homeomorphism.

One often desires more structure on a manifold than simply the topological structure. For example, if one would like an unambiguous notion ofdifferentiation of functions on a manifold, then it is necessary to construct an atlas whose transition functions aredifferentiable. Such a manifold is calleddifferentiable. Given a differentiable manifold, one can unambiguously define the notion oftangent vectors and thendirectional derivatives.

If each transition function is asmooth map, then the atlas is called asmooth atlas, and the manifold itself is calledsmooth. Alternatively, one could require that the transition maps have onlyk continuous derivatives in which case the atlas is said to be${\displaystyle C^k}$.

Very generally, if each transition function belongs to apseudogroup${\displaystyle \mathcal{G}}$ of homeomorphisms of Euclidean space, then the atlas is called a${\displaystyle \mathcal{G}}$-atlas. If the transition maps between charts of an atlas preserve alocal trivialization, then the atlas defines the structure of a fibre bundle.

• Smooth atlas
• Smooth frame

• Atlas by Rowland, Todd

• Manifolds

• CS1 German-language sources (de)
• Articles with short description
• Short description is different from Wikidata
• Wikipedia articles needing clarification from May 2024