Lattices and manifolds of classes of flat tori

Abstract

The topological and differentiable structures of some moduli spaces constructed from flat Riemannian tori (concretely, H\(Gl(2,R)+/Sl(2, Z)), H = O+(2,R), CO+(2,R), O(2,R), CO(2,R)) is studied by means of a cut-and-paste procedure. In the orientation preserving cases, the quotients admit a structure of orbifold with singular points corresponding to lattices in the square and hexagonal crystal systems. In the non-orientation preserving ones, the structure is a smooth manifold with piecewise smooth boundary, where the interior points corresponds to oblique lattices, and the edge of the boundary to square and hexagonal ones.