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.
