Almost Flat Manifolds

This thesis is based on the course " Convergence and Collapsing Theory in Riemanian Geometry " Prof. Rong taught in the spring semester of 2006 at Capital Normal University. The author first briefly review basic definitions and theorems related to this thesis and mainly focus on the whole proof of Gromov's Almost Flat Manifolds Theorem.M.Gromov([5]) first proved that any almost flat manifold has a finite covering which is diffeomorphic to nilpotent Lie group modula its discrete subgroup. P.Buser and H.Karcher([7]) gave the more details of this proof later. Then E.A.Ruh([6]) proved a stonger version of above theorem: any almost flat manifold is diffeomorphic to an infra-nihnanifold. Prof. Rong proposed a new proof in his unpublished lecture notes , but still had the part" of flat connection with parallel torsion unfinished. In this thesis the author will give the remained part of the proof which uses the methods of E.A.Rub and Ghanaat. The author will also give the details of Prof. Rong's proof.