Cut Loci and Diameters of the Berger Lens Spaces

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, which has already been 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 $L(p; 1)$.