On Holonomy Covering Spaces

Introduction. In [l] L. Markus and the author introduced the concept of holonomy covering spaces for flat affinely connected manifolds or what are also called locally affine spaces. We also proved in [l ] that the holonomy covering space of a complete « dimensional locally affine space must be some « dimensional cylinder, i.e., T^XF"-*, i = 0, •■-,«, where T* denotes the locally affine i dimensional torus and £"_i denotes the «—i dimensional affine space. It is the purpose of this paper to prove that all these holonomy covering spaces are actually realized. To be more explicit, we shall construct n locally affine connections A", i=l, •••,«, on the « dimensional torus such that the holonomy covering space of A" is TiXFn~i, i = l, • • • , «.