Distality, completeness, and affine structures (Q1085808)

An affine connection \(\nabla\) on a smooth manifold M is called distal if the holonomy group \(G_ m\subset Hom(T_ mM)\) is a distal linear group for each \(m\in M\). The author proves that a closed affine manifold with distal holonomy is complete. Then, he treats a characterization result. Theorem. ''Let M be a closed affine manifold. M has distal holonomy if, and only if, some finite cover \(M_ 0\) of M is a nilmanifold \(\Gamma\setminus N\) with the affine structure induced from a complete left invariant affine structure on the simply connected nilpotent Lie group N''. - A left invariant complete affine structure on a simply connected Lie group N is the same as a simply transitive affine action of N on an affine space A. It was conjectured that such an action contains a nonzero translation if N is nilpotent [cf. \textit{L. Auslander}, Am. J. Math. 99, 809-826 (1977; Zbl 0357.22006)]. While this is true in dimensions \(<4\), the author gives an example that proves the conjecture to be false in general. He also shows, by imposing a further strong additional assumption, that a variant of the conjecture holds true. Finally, he proves a result on ergodic automorphisms as acting on a nilmanifold N.