altering curvature on a tessellation representation of a compact surface - MathOverflow most recent 30 from http://mathoverflow.net 2013-06-18T07:13:22Z http://mathoverflow.net/feeds/question/34694 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/34694/altering-curvature-on-a-tessellation-representation-of-a-compact-surface altering curvature on a tessellation representation of a compact surface Nicolas Fernandez-Arias 2010-08-05T21:55:25Z 2010-08-22T20:34:42Z <p>I have been reading about tessellation representations of compact surfaces, such as how the square tiling the plane represents the torus. For surfaces of genus > 1 (the ones that interest me), we need to move to the hyperbolic plane in order to obtain a tessellation representation, which makes sense because of Gauss-Bonnet's restriction that the total curvature of a surface of genus > 1 be negative. </p> <p>But, since we had to move to the hyperbolic plane in order to obtain a tessellation, my question is this: can we only find tessellation representations of compact surfaces of genus > 1 when we impose constant negative curvature on them? Or does the tessellation representation look the same when I change curvature on certain sets? Or maybe, is it slightly perturbed but still a tessellation? If my questions are completely misguided, could someone suggest some literature? Thank you!</p> http://mathoverflow.net/questions/34694/altering-curvature-on-a-tessellation-representation-of-a-compact-surface/36395#36395 Answer by Sam Nead for altering curvature on a tessellation representation of a compact surface Sam Nead 2010-08-22T20:34:42Z 2010-08-22T20:34:42Z <p>It is always easier to think about the spherical case first. You can see that all round, two-dimensional spheres, regardless of radius, admit tessellations. So constant curvature \$+1\$ is not necessary. On the other hand, constant positive curvature should be a requirement. (In fact, it is fair to require that the metric be homogeneous and isotropic. Otherwise what does it mean for tiles of the tessellation to be identical?)</p> <p>The case of constant negative curvature is the same. The choice of constant doesn't really matter, so you might as well use \$-1\$. For an elementary discussion, with many beautiful pictures, see "Noneuclidean tesselations and their groups", by Wilhelm Magnus. A more modern treatment, also with wonderful graphics is "Indra's Pearls" by Mumford, Series, and Wright. </p>