A. V. Podobryaev, Antipodal Points and Diameter of a Sphere

Title: Antipodal Points and Diameter of a Sphere
Authors: A. V. Podobryaev
Journal title: Russian Journal of Nonlinear Dynamics
Year: 2018
Issue: 4
Volume: 14
Pages: 579–581

A. V. Podobryaev, “Antipodal Points and Diameter of a Sphere”, Russian Journal of Nonlinear Dynamics, 14:4 (2018), 579–581


We give an example of a Riemannian manifold homeomorphic to a sphere such that its diameter cannot be realized as a distance between antipodal points. We consider a Berger sphere, i.e., a three-dimensional sphere with Riemannian metric that is compressed along the fibers of the Hopf fibration. We give a condition for a Berger sphere to have the desired property. We use our previous results on a cut locus of Berger spheres obtained by the method from geometric control theory.