Coordinate Systems

Long overdue post to the Poincaré Project.

When we defined manifolds back in Poincare Project: Manifolds, we pretty much defined what a coordinate system is:

Define a chart to be a continuous one-to-one mapping from an open set to R^n.

A manifold is a topological space covered by one or more charts. In other words, every point (and some open set the point is in) is part of at least one chart.

A chart provides a coordinate system and the coordinates of a point on a manifold are just the components of the point in R^n that the point on the manifold maps to in that chart.

So one way of thinking about coordinates is that they are tuples of real numbers that identify points on a manifold.

But it is important to note that coordinate systems are continuous. If you range one coordinate over a subset of its possible values, keeping any other coordinates constant, then you won't just get a random set of points, you'll get a continuous curve.

Note also that, although we often think about latitude and longitude as providing a coordinate system for the two-dimensional sphere, this single coordinate system breaks down at the poles. The north and south poles don't have a defined longitude. This doesn't pose a problem for either the latitude / longitude coordinate system or the manifold-ness of the sphere because (a) the latitude / longitude system works fine everywhere but the poles; (b) there is nothing wrong with a manifold requiring multiple overlapping coordinate systems to cover all its points.

To make this clearer, consider an even simpler example: the one-dimensional sphere, or circle. We could just map the points on the circle to [0, 1) but then we would get a discontinuity back at 0. So we have to use multiple overlapping coordinate systems:

Here we have a red coordinate system which covers part of the (black) circle and a blue coordinate system which covers a different but overlapping part. For every point, there are coordinates for that point in at least one system and the system is continuous at that point. I've arbitrarily ranged the two coordinate systems from 0 to 1 but any continuous parameterisation of the coordinate curve would work.

Next we will reintroduce the notion of a metric defined in terms of coordinate systems.

UPDATE: next post