Introduction. The sphere is known to be the universal covering for complete connected Riemannian manifolds of constant positive curvature. More precisely, if M is an n-dimensional complete connected Riemannian manifold of constant sectional curvature k2 > 0 with k > 0, and if Sn is the sphere of radius k-1 in Euclidean space RI'+', with the induced metric, then there is a covering of M by Sn such that the covering projection is a local isometry. Because of this phenonmenon, the complete connected Riemannian manifolds of constant positive curvature are called the "spherical spaceforms." In his thesis, G. Vincent [14] attempted to classify them. Following this line of investigation, we take a compact connected Riemannian homogeneous manifold M and ask which Riemannian manifolds admit M as a Riemannian covering manifold. In Chapter I, this problem is reduced to a problem on discrete subgroups of compact Lie groups: