Collapsing Riemannian manifolds to ones of lower dimensions

In Several authors found that interesting phenomena occur when a sequence of Riemannian manifolds f, collapses to a lower dimensional space X. (Examples of such phenomena will be given later.) But, in general, it seems very difficult to describe the relation between topological structures of M t and X. In this paper, we shall study the case when the limit space X is a Riemannian manifold and the sectional curvatures of M i are bounded, and shall prove that, in that case, M, is a fiber bundle over X and the fiber is an infranilmanifold. Here a manifold F is said to be an infranilmanifold if a finite covering of F is diffeomorphic to a quotient of a nilpotent Lie group by its lattice.