What is an Atlas & how do I use it?

As the name suggests, an atlas corresponds to a collection of maps, each of which shows a piece of a manifold and looks like flat Euclidean space. To use an atlas, one needs to know how the maps overlap. To be useful, the maps must not be too different on these overlapping areas.

What is a smooth atlas?

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 .

What is the plural of Atlas?

The plural of atlas is atlases, although some authors use atlantes. An atlas on an -dimensional manifold is called an adequate atlas if the image of each chart is either or , is a locally finite open cover of , and , where is the open ball of radius 1 centered at the origin and is the closed half space.

Which chart is traditionally recorded as an ordered pair?

The chart is traditionally recorded as the ordered pair . An atlas for a topological space is an indexed family of charts on which covers (that is, ).