is complete if and only if it has parallel volume. Known in dimension 2. Auslander conjecture (1964) stating that any affine crystallographic group contains a polycyclic subgroup of finite index. Known in dimensions up to 6, and when the holonomy of the flat connection preserves a Lorentz metric. Since every virtually polycyclic crystallographic group preserves a volume form, Auslander conjecture implies the "only if" part of the Markus conjecture. Chern conjecture (1955) The Euler class of an affine manifold vanishes. Notes References Group theory Affine geometry Structures on manifolds Differential geometry Manifolds