Antipodal Points and Diameter of a Sphere

Abstract

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.

On the diameter of odd-dimensional spheres

“…Podobryaev [Po18b] found counterexamples for n = 3 as a consequence of the explicit expression for the diameter of every 3-dimensional Berger sphere established by himself in [Po18a] (see also [PS16]).…”

Diameter and displacement of sphere involutions

“…Podobryaev [Po18b] observed that sufficiently collapsed Berger spheres provide a negative answer in dimension n = 3. In fact, this observation can be easily extended to all odd dimensions n ≥ 3, considering the (homogeneous) spheres (S 2q+1 , g(t)) obtained scaling the unit round sphere by t > 0 in the vertical direction of the Hopf bundle S 1 → S 2q+1 → CP q .…”