Affine manifold

In differential geometry, an affine manifold is a differentiable manifold equipped with a flat, torsion-free connection.
Equivalently, it is a manifold that is (if connected) covered by an open subset of formula_1, with monodromy acting by affine transformations. This equivalence is an easy corollary of Cartan–Ambrose–Hicks theorem.
Equivalently, it is a manifold equipped with an atlas—called the affine structure—such that all transition functions between charts are affine transformations (that is, have constant jacobian matrix); two atlases are equivalent if the manifold admits an atlas subjugated to both, with transitions from both atlases to a smaller atlas being affine. A manifold having a distinguished affine structure is called an affine manifold and the charts which are affinely related to those of the affine structure are called affine charts. In each affine coordinate domain the coordinate vector fields form a parallelization of that domain, so there is an associated connection on each domain. These locally defined connections are the same on overlapping parts, so there is a unique connection associated with an affine structure. Note there is a link between linear connection (also called affine connection) and a web.
Formal definition.
An affine manifold formula_2 is a real manifold with charts formula_3 such that formula_4 for all formula_5 where formula_6 denotes the Lie group of affine transformations. In fancier words it is a (G,X)-manifold where formula_7 and formula_8 is the group of affine transformations. 
An affine manifold is called complete if its universal covering is homeomorphic to formula_1.
In the case of a compact affine manifold formula_10, let formula_8 be the fundamental group of formula_10 and formula_13 be its universal cover. One can show that each formula_14-dimensional affine manifold comes with a developing map formula_15, and a homomorphism formula_16, such that formula_17 is an immersion and equivariant with respect to formula_18.
A fundamental group of a compact complete flat affine manifold is called an affine crystallographic group. Classification of affine crystallographic groups is a difficult problem, far from being solved. The Riemannian crystallographic groups (also known as Bieberbach groups) were classified by Ludwig Bieberbach, answering a question posed by David Hilbert. In his work on Hilbert's 18-th problem, Bieberbach proved that any Riemannian crystallographic group contains an abelian subgroup of finite index.
Important longstanding conjectures.
Geometry of affine manifolds is essentially a network of longstanding conjectures; most of them proven in low dimension and some other special cases.
The most important of them are: