Building a coordinate system on a surface - Part 4
(Or Math of GR part 13 for Calculus nerds)

The interesting part, which is where Mathematician Riemann gets involved, is to see if the math works out when we consider the surface S as its own two-dimensional mathematical space. As in real space, we can create coordinate system that can map the location of points, but note what we actually have done. We located the position of a point on a surface by a coordinate system that was ON the surface.

Can we continue to do this and "pretend" that our surface is not an object embedded in a larger 3D space, that it is a 2D Space that can stand as such all on its own? So far, the answer is yes. Not only did Riemann prove that this idea was mathematically consistent, but Einstein applied the mathematical modeling to Gravity, and was able to develop the most successful Physical Theory in Science that was ever discovered.

More next.