JournalsjemsVol. 5 , No. 3DOI 10.1007/s10097-003-0051-7

Upper bounds on the length of a shortest closed geodesic and quantitative Hurewicz theorem

  • Alexander Nabutovsky

    University of Toronto, Canada
  • Regina Rotman

    University of Toronto, Canada
Upper bounds on the length of a shortest closed geodesic and quantitative Hurewicz theorem cover

Abstract

In this paper we present two upper bounds on the length of a shortest closed geodesic on compact Riemannian manifolds. The first upper bound depends on an upper bound on sectional curvature and an upper bound on the volume of the manifold. The second upper bound will be given in terms of a lower bound on sectional curvature, an upper bound on the diameter and a lower bound on the volume. The related questions that will also be studied are the following: given a contractible k-dimensional sphere in Mn, how fast can this sphere be contracted to a point, if pgri(Mn)={0} for 1lei