Distances determine shape

I was trying to understand why the metric matters, by reading the black bible of gravitation. I came across this application of the Gauss-Bonnet Theorem on a tetrahedron, that if at each of the four vertices we lay out the equilateral triangles on a flat plane, the angles will fail to meet by pi. Summing them up, we have a total failure of 4pi, which is robust to deformations that don't change the topology of the two-sphere. I must remember these simple applications! When it comes to convincing people (myself), simplicity is strength. And finally I have a more kick-ass introduction to topology than the lame-ass ones I always give friends and family.