The idea behind the geometrical description of a closed curved space

Question

This question is about the idea of how to describe a curved space, I'm not asking for formulas because I've never studied this topic before.
Let's imagine an ant on a sphere, it will use two coordinates to describe the space because it doesn't see the third dimension. How the ant will set up the coordinate system and describe the world it sees? I think it will use two axis that are perpendicular as humans do with longitude and latitude for earth and then making observation it will derive different geometrical rules respect to the euclidean geometry. But in this way it can't describe the north and the south pole because for example there are many longitude that correspond to the North Pole. Can you explain me the idea behind the description of a closed curved space?

Answer 1

You are perfectly right, you can't always describe a curved space (a Riemanian - or pseudo-Riemanian for space-time - manifold) with only one set of coordinates, and it is not required. It's OK as long as you can use several that "nicely overlap". Here for example, a map from North Pole, where South Pole becomes a line (is "degenerated"), from which you exclude the South polar region, and one anywhere else, for example around South Pole, where you also exclude the North Polar region. On the overlap, they will describe the world in the same way, so you can navigate anywhere using this set of two maps, and so the manifold is nice enough to be worked with.

Notice that in each of these "coordinate patches", each point from the sphere is represented only once, which ensures that everything is OK and is, therefore, a requirement.