On holonomy covering spaces

Abstract

Introduction. In [1 ] 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 [1 ] that the holonomy covering space of a complete n dimensional locally affine space must be some n dimensional cylinder, i.e., TiXFn-i i=0, *... , n, where Ti denotes the locally affine i dimensional torus and En-i denotes the n-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 =1, * , n, on the n dimensional torus such that the holonomy covering space of A' is T4XFn-!, il***, n.