Geodesics on a Flat Torus Projected to a Standard Torus in 3D
I'm only uploading this to YouTube because it's too big to convert into a gif to put on Medium. This path is what you'd get if you were to travel in a straight line in normal 2D Euclidean space in a square, but you were teleported to the opposite side every time you hit one of the sides. If you then glue the sides together to get rid of the teleportation, you get a torus. To glue it in 3D normally, you have to stretch the torus so that you travel farther when you go a certain angle on the outside, which is why it seems to speed up and slow down at different parts even though it should be going at a constant speed for a geodesic.
In other words, the whole point of this video is to show that two Riemannian manifolds can refer to the same manifold, but have different properties because of the different metric tensors.