Vol. 94, No. 2, 1981

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Two examples of affine manifolds

William Goldman

Vol. 94 (1981), No. 2, 327–330

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 ΩΓ 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.

Mathematical Subject Classification 2000
Primary: 57R15
Secondary: 53C30
Received: 11 December 1979
Published: 1 June 1981
William Goldman
Department of Mathematics
University of Maryland
College Park MD 20742
United States