I'm doing a project involving tilings of Minkowski space. For instance in 2d I have rectangular tiles determined by a spacelike line segment: the rectangle is the region caused by the line segment. Four of these fit together at a vertex and there is a condition on the lengths of the four spacelike lines that determines whether they fit together properly in Minkowski space (namely: the product of the left and right lengths equals the product of the top and bottom lengths).
If you assemble tiles into closed manifolds using local neighborhood rules such as these, you get other things besides Minkowski space, globally. Thinking of each vertex with its four rectangles that meet there as a chart, we have: each chart is a map of a region of the manifold to a region of Minkowski space, and the charts are compatible in that the metric is preserved on overlaps. What are manifolds like this called? (They are pseudo-Riemannian manifolds in which there is a compact region about each point that is isometric to a region of Minkowski space. So they're flat, for starters...)
