The Earth is a classic example of a 2D manifold. Looks Euclidean to us, but is most definitely curved.
I am self teaching some differential geometry and I don't quite understand the difference between two things.
So I understand that a chart is a mapping between a subset of the manifold and Eulclidean space, but what I don't understand is how this relates to maps of the Earth.
I have read that it is impossible to represent the surface of the Earth/sphere with just a single chart. However, there are definitely maps of the earth that encompass every point. I have seen one that is centered around the north pole, for example, that includes the whole surface – except the south pole is definitely misrepresented somewhat, as it is wrapped around the edges.
So what is it about map projections that means they are not considered charts?
One presumes some level of maths is needed to perform a map projection, are there specific rules for what makes a chart a chart? And presumably these rules are broken for map projections?
Best Answer
A mapping $U \to \mathbb R^n$ from an open subset $U$ of your manifold is not a chart if one point of $U$ is "wrapped around the edges."
You may have answered your own question with that single sentence.
The problem isn't that it's impossible to make a map projection covering every point on Earth--the problem is that every such projection represents at least one point too many times. It gives a one-to-many correspondence that is not a function. Alternatively, if you erase the redundant copies of each such point and keep just one, you still don't have a continuous mapping at that point, so you still don't have a chart in the mathematical sense.
If you remove one point from a sphere $S$ to get an open set $S_1,$ a stereographic projection gives a nice conformal mapping $S_1 \to \mathbb R^2.$ In mathematics, however, a sphere with one point removed is not a sphere. It's a sphere with one point removed.