Atlas (topology)

Charts{{anchor|Maps}}

The definition of an atlas depends on the notion of a chart. A chart for a topological space M 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).

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

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_{\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 is said to be an n-dimensional manifold.

The plural of atlas is atlases, although some authors use atlantes.

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. 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_i\right)_{i \in I} is a refinement of \mathcal{V}.

Transition maps

| image-width = 250 } } 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 \tau_{\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.

References

• , Chapter 5 "Local coordinate description of fibre bundles".

References

1. (2005). "Vektoranalysis". Springer.
2. Jost, Jürgen. (11 November 2013). "Riemannian Geometry and Geometric Analysis". Springer Science & Business Media.
3. (9 March 2013). "Calculus of Variations II". Springer Science & Business Media.
4. Kosinski, Antoni. (2007). "Differential manifolds". Dover Publications.