Two Examples of Affine Manifolds

An affine manifold is a manifold with a distinguished system of affine coordinates, namely, an open covering by charts which map homeomorphically onto open sets in an affine space E such that on overlapping charts the homeomorphisms differ by an affine automorphism of E. Some, but certainly not all, affine manifolds arise as quotients Ω/Γ of a domain in E by a discrete group Γ of affine transformations acting properly and freely. In that case we identify Ω with a covering space of the affine manifold. If Ω—E, then we say the affine manifold is complete. In general, however, there is only a local homeomorphism of the universal covering into E, which is equivariant with respect to a certain affine representation of the fundamental group. The image of this representation is a certain subgroup of the affine group on E, is called the affine holonomy and is well defined up to conjugacy in the affine group. See Fried, Goldman, and Hirsch.