Coordinate Chart

A coordinate chart is a way of expressing the points of a small neighborhood, usually on a manifold , as coordinates in Euclidean space. An example from geography is the coordinate chart given by the functions of latitude and longitude. This coordinate chart is not valid on the whole globe, since it doesn't give unique coordinates at the north or south pole (which way is east from the north pole?).

Technically, a coordinate chart is a map

where  is an open set in ,  is an open set in  and  is the dimension of the manifold. Often, through notational abuse, the open set  is equated with , and calculations on the manifold are done in the coordinate chart. This technique has the drawback that it must be checked whether a change of coordinates affects the result of a calculation.

The map  must be one-to-one, and in fact must be a Homeomorphism. On a smooth manifold, it must be a diffeomorphism, although if the chart defines the smooth structure then this is a tautology. Similarly, on a complex manifold, the map  is holomorphic.

If there are two neighborhoods  and  with coordinate charts  and , the transition function  is well-defined since coordinate charts are one-to-one.