Locally Cartesian Coordinates (on a torus)

The image below shows a small portion of a coordinate net on a torus. This particular net is geodesic at the red point, and this means that the infinitesimal rule for calculating distances is the Pythagorean theorem. To put it another way, very small two dimesional inhabitants of the torus (who live near the red point, who have limited measuring capabilities, and who we might call surfers) will think that their space is a flat Euclidean plane.

In the image shown below, the coordinate net has been greatly expanded to the extent that it covers a significant portion of the torus. When the surfers travel to the far corners of the net, they will discover that the infinitesimal rule of distance has become the law of cosines. This suggests to them, but does not prove, that their world is curved: In order to demonstrate that their world is flat, one way in which they could proceed would be to exhibit a coordinate system which covers their entire world and in which the infinitesimal rule of distance is the Pythagorean theorem at every point. The failure of the given coordinate system to meet this condition only indicates that the space has not been shown to be Euclidean. The question as to whether the space is flat or curved is still open.

In the first image below, the surfers have travelled along a geodesic beginning at the red point and coming back to the red point. The light blue line was parallel propagated along the way and apparently ends up the same as when it started. This last fact would be a property of a flat space, if it were not for the fact that a geodesic path was followed and returned to the starting point. In the second image below a slightly different path, which is not a geodesic, was followed with the result that the starting and ending blue lines are different. From this the surfers conclude that their space has curvature somewhere. In a future test, the surfers will attempt to demonstrate that their land is curved at the red point. They will do this by looking at a "blue line" parallel propagated around a small closed loop which lies completely within the gray grid shown below. The path will pass through the red point, and the surfers will undertake a limit process in which the loop shrinks to zero area. If they can detect that the line changes its direction when it returns to its starting point, then they will know that their space is curved at the red point.