Affine manifolds with dilations

An affine manifold is called a manifold with dilations if its holonomy is contained in a group of the form G = N ⋊ (A × K) ⊂ Aff(Eℓ), where N is a nilpotent group acting simply transitively on Eℓ; K is a compact subgroup, and A is a 1-parameter subgroup of “dilations”. e (Mℓ,G) is a compact connected affine manifold with dilations of dimension ⩾ 2. Assume that the holonomy group acts on the image of a development map as a covering transformation. We prove that: If M is geodesically incomplete, then it is finitely covered by a Hopf manifold, (Eℓ − 0)Z ≅ S1 × Sℓ − 1.