Atlas (topology)

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In mathematics, particularly topology, an atlas describes how a manifold is equipped with a differential structure. Each piece is given by a chart (also known as coordinate chart or local coordinate system).

The definition of an atlas depends on the notion of a "chart". In topology, a chart on a manifold M is defined to be a homeomorphism φ from an open subset U of M to an open subset V of \mathbb{R}^n. If (U_{\alpha}, \varphi_{\alpha}) and (U_{\beta}, \varphi_{\beta}) are two charts on M such that U_{\alpha} \cap U_{\beta} is non-empty, then define the transition map

Note that since \varphi_{\alpha} and \varphi_{\beta} are both homeomorphisms, the transition maps are also homeomorphisms. So, the transition maps are already endowed with a kind of compatibility in the sense that changing from the coordinate system on one chart to the coordinate system on another chart is continuous.

Then an atlas on a manifold M is a collection \mathcal{A} = \{(U_{\alpha}, \varphi_{\alpha})\} of charts on M whose domains cover M.

Two overlapping charts (U_{\alpha}, \varphi_{\alpha}) and (U_{\beta}, \varphi_{\beta}) are smoothly compatible if the transition map between them is infinitely differentiable as a map from Euclidean space to itself.

Having defined these notions, a smooth atlas on M is an atlas where we make the additional requirement that, for any two overlapping charts on M, the transition maps between them are smoothly compatible.

Two atlases \mathcal{A} and \mathcal{B} on M are smoothly compatible if all charts in \mathcal{A} which overlap charts in \mathcal{B} are smoothly compatible. If this is the case then \mathcal{A} \cup \mathcal{B} is also a smooth atlas on M. This gives a natural equivalence relation, from which we can consider an equivalence class of smoothly compatible atlases, which we call the maximal atlas. A manifold M together with a maximal atlas is said to have a smooth structure. There are, in higher dimensions, examples of topological manifolds with multiple different smooth structures. One of the first examples was John Milnor's discovery of an exotic sphere, a 7-manifold which is homeomorphic to the 7-sphere but not diffeomorphic.

In general, doing computations with the maximal atlas of a manifold is unwieldy and we need only choose one particular smooth atlas to work with. Maximal atlases are needed for the unambiguous definition of smooth maps from one manifold to another.

The differentiability requirements on the transition functions can be weakened, so that we only require the transition maps to be k-times continuously differentiable; or strengthened, so that we require the transition maps to real-analytic. Accordingly, this gives a Ck or analytic structure on the manifold rather than a smooth one. Similarly, we can define a complex manifold by requiring the transition maps to be holomorphic.

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