An affine manifold is a topological manifold which admits a system of charts such that the coordinate changes are (restrictions of) affine transformations. Let $M$ be a compact affine manifold. Let $G$ be the fundamental group of $M$ and $\tilde M$ be its universal cover. One can show that each $n$-dimensional affine manifold comes with a developing map $D\colon \tilde M \to \mathbb R^n$, and a homomorphism $\varphi \colon G \to {\rm Aff}(\mathbb R^n)$, such that $D$ is an immersion and equivariant with respect to $\varphi$.

An affine manifold is called *complete* if $D$ is a homeomorphism, in this case: $\varphi$ is injective, $G$ is a Bieberbach group, and $M$ is aspherical, i.e. $\tilde M$ is contractible. The non-complete case seems to be far more complicated.

Question 1:Is there an easy example, where $D$ is not surjective?

Question 2:Is there an easy example, where $\varphi$ is not injective?

Question 3:Is there an easy example, where $M$ is not aspherical?

**EDIT:** As André suggested, let's ask for examples for which $\varphi$ takes values in $SL(n,\mathbb R) \ltimes \mathbb R^n$ or even $SL(n,\mathbb Z) \ltimes \mathbb R^n$, seen as subgroups of ${\rm Aff}(\mathbb R^n)$.