Complete affine manifolds: a survey

An affinely flat manifold (or just affine manifold) is a manifold with a distinguished coordinate atlas with locally affine coordinate changes. Equivalently M is a manifold equipped with an affine connection with vanishing curvature and torsion. A complete affine manifold M is a quotient E/Γ where Γ ⊂ Aff(E) is a discrete group of affine transformations acting properly on E. This is equivalent to geodesic completness of the connection. In this case, the universal covering space of M is affinely diffeomorphic to E, and the group π1(M) of deck transformations identifies with the affine holonomy group Γ. Flat Riemannian manifolds are special cases where Γ is a group of Euclidean isometries. The classical theorems of Bieberbach provide a very satisfactory picture of such structures: every compact flat Riemannian manifold is finitely covered by a flat torus E/Λ where Λ ⊂ G is a lattice in the group G of translations of E. Furthermore every complete flat Riemannian manifold is a flat orthogonal vector bundle over its a soul, a totally geodesic flat Riemannian manifold. (See, for example, Wolf [29].) An immediate consequence is χ(M) = 0 if M is compact (or even if Γ is just nontrivial). This follows immediately from the intrinsic Gauß-Bonnet theorem of Chern [11], who conjectured that the Euler characteristic of a closed affine manifold vanishes. (Chern-Gauß-Bonnet applies only to orthogonal connections and not to linear connections.) In this generality, Chern’s conjecture remains unsolved. Affine manifolds are considerably more complicated than Riemannian manifolds, where metric completeness is equivalent to geodesic completeness. In particular, simple examples such as a Hopf manifold R \ {0}/〈γ〉, where γ is a linear expansion of R illustrate that closed affine manifolds need not be complete. For this reason we restrict only to geodesically complete manifolds. Kostant and Sullivan [20] proved Chern’s conjecture when M is complete. In other directions, Milnor [24] found flat oriented R-bundles over surfaces with nonzero Euler class. Using Milnor’s examples, Smillie [26] constructed flat affine connections on some manifolds of nonzero Euler characteristic. (Although the curvature vanishes, it seems hard to control the torsion.) Auslander’s flawed proof [4] of Kostant-Sullivan still contains interesting ideas. Auslander claimed that every closed complete affine manifold is finitely covered by a complete affine solvmanifold G/Γ, where G ⊂ Aff(E) is (necessarily solvable) closed subgroup of affine automorphisms of E. This generalizes Bieberbach’s structure theorem for flat Riemannian manifolds. Whether every closed complete affine manifold has this form is a fundamental question in its own right, and this question is now known as the “Auslander Conjecture.” ([16]). It has now been established in all dimensions n < 7 by Abels-Margulis-Soifer [2, 3]. Milnor’s paper [25] clarified the situation. Influenced by Tits [28] he asked whether any discrete subgroup of Aff(E) which acts properly on E must be virtually polycyclic. If so then complete affine manifolds admit a simple structure, and can be classified by techniques similar to the Bieberbach theorems. Tits’s theorem