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.



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> **Question 1:** Is there an easy example, where $D$ is not surjective? 
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> **Question 2:** Is there an easy example, where $\varphi$ is not injective? 
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> **Question 3:** Is there an easy example, where $M$ is not aspherical? 
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**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 ${\rm Aff}(\mathbb R^n)$.



Affine manifolds
There is a conjecture due to Markus which states that any compact affine manifold has parallel volume (*i.e.* the linear part of $\varphi$ lies in $\mathrm{SL}(n;\mathbb{R})$) if and only if it is complete. To the best of my knowledge, this conjecture is still open, which goes towards saying that there should be no easy examples to questions 1 and 3 for affine manifolds with parallel volume. 


If the fundamental group $G$ of a compact affine manifold with parallel volume is nilpotent, the beautiful *Affine manifolds with nilpotent holonomy* by Fried, Goldman and Hirsch, Comm. Math. Helv. **56** (1981) proves that Markus' conjecture holds in this case and, thus, there are no examples to questions 1 and 3 with nilpotent fundamental group. The proof is a cunning mixture of representation theory and geometry, so I strongly recommend taking a look at it. The results in this paper also imply that the above examples (to questions 1 and 3) constructed by Andre Henriques cannot be adapted so that the resulting manifolds admit parallel volume (these are, nonetheless, very nice examples to the original question!). 


