Collapsing Riemannian manifolds to ones of lower dimensions

Abstract

IntroductionIn [7], Gromov introduced a notion, Hausdorff distance, between two metric spaces.Several authors found that interesting phenomena occur when a sequence of Riemannian manifolds Λf, 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.A complete Riemannian manifold M is contained in class Jί(n) if dim M < n and if the sectional curvature of M is smaller than 1 and greater than -1.An element N of Jt{n) is contained in Jί(n,μ) if the injectivity radius of N is everywhere greater than μ.Main Theorem.There exists a positive number ε(n,μ) depending only on n and μ such that the following holds.IfMeJP(n), N E:J({n,μ), and if the Hausdorff distance ε between them is smaller than ε(«, μ), then there exists a map f:M -> N satisfying the conditions below.(0-1-1) (M,NJ) is a fiber bundle.(0-1-2) The fiber offis diffeomorphic to an infranilmanifold.(0-1-3) Ifξ^ T{M) is perpendicular to a fiber off, then we have e-Ύ{t) <\df(ί)\/\ί\ <e τ{ε \

Locations

• Journal of Differential Geometry - View - PDF