Cut loci and diameters of the Berger lens spaces

TL;DR

This paper investigates the geometry of deformed Riemannian metrics on 3D lens spaces, analyzing their cut loci, diameters, and convergence to sub-Riemannian structures using geometric control theory.

Contribution

It introduces a one-parameter family of axisymmetric metrics on lens spaces, computes their cut loci and times, and establishes diameter bounds matching known values.

Findings

Cut loci and cut times converge to those of the sub-Riemannian limit.

Derived lower bounds for diameters that match exact values for L(p;1).

Methods from geometric control theory are effectively applied.

Abstract

In this paper, we study Riemannian metrics on the three-dimensional lens spaces that are deformations of the standard Riemannian metric along the fibers of the Hopf fibration. In other words, these metrics are axisymmetric. There is a one-parametric family of such metrics. This family tends to an axisymmetric sub-Riemannian metric. We find the cut loci and the cut times using methods from geometric control theory. It turns out that the cut loci and the cut times converge to the cut locus and the cut time for the sub-Riemannian structure, that was already studied. Moreover, we get some lower bounds for the diameter of these Riemannian metrics. These bounds coincide with the exact values of diameters for the lens spaces L(p;1).