An affine manifold (M,C), is a differentiable manifold M endowed with a connection C which curvature and torsion forms vanish identically. The connection defines on M an atlas which transitions maps are affine maps. The affine manifold is complete if and only if the connection is complete. 40 years ago, L. Auslander has proposed the following conjecture: The fundamental group of a compact and complete affine manifold is polycyclic. In [1] I have proposed the following conjecture: Let (M,C) be a compact and complete affine manifold, there exists a finite galoisian cover of (M,C) which is the source space of a non trivial affine map. This last conjecture implies the Auslander conjecture and leads to the following problem: Given two compact and complete affine manifolds (B,C_1) and (F,C_2) classify every affine bundle which base space is (B,C_1) and which typical fiber is diffeomorphic to F and is endowed with an affine structure which linear holonomy is the one of (F,C_2). The purpose of my talk is to present this problem and to show how I used stack theory to solve it.
[1] Tsemo, A. Dynamic of affine manifolds J. London Math. Society (63)2001 469-486