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Cut loci and diameters of the Berger lens spaces

Alexey Podobryaev

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).

Cut loci and diameters of the Berger lens spaces

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).

Paper Structure

This paper contains 7 sections, 14 theorems, 49 equations, 6 figures.

Table of Contents

  1. Introduction
  2. Axisymmetric Riemannian metrics on lens spaces
  3. Some necessary definitions and known results
  4. Maxwell strata
  5. The Maxwell time is less or equal to the conjugate time
  6. The cut locus and the cut time
  7. Lower bound for diameter

Key Result

Proposition 1

Assume that $p > 1$. Consider the subset Let us glue the points $(q_1,q_2,q_3) \sim (q_1',q_2',q_3')$ of the boundary $\partial L$ if one of the following conditions holds: (1) $q_3 > 0$, $q_3' = -q_3$ and $z' = q_1' + iq_2' = e^{\frac{2\pi q}{p}}z$, where $z = q_1 + iq_2$, (2) $q_3 = q_3' = 0$, and $z' = e^{\frac{2\pi k}{p}}z$ for some $k

Figures (6)

  • Figure 1: The lens space $L(p;q)$, $p \geqslant 2$ as a domain $L \subset \mathbb{R}^3$ with identified points on its boundary $\partial L$. Identified points have the same marks. The general view (left) and the projection to the $(q_1,q_2)$-plane (right) for $p = 8$ and $q = 3$.
  • Figure 2: The functions $\tau_{\ell}^-, \tau_{\ell}^+ : [0,1] \rightarrow \mathbb{R}_+$.
  • Figure 3: The function $\tau_{\ell}^-$ and $\pi$ in the case $\eta < -\frac{p-1}{p}$.
  • Figure 4: The cut locus for axisymmetric Riemannian metric on the lens space $L(p;q)$ for $p > 1$ and $\eta < -\frac{p-1}{p}$ has two strata $\partial L / \sim$ and an interval.
  • Figure 5: The cut locus for axisymmetric sub-Riemannian metric on the lens space $L(p;q)$ for $p > 1$ has two strata $\partial L / \sim$ and a punctured circle. This corresponds to $\eta \rightarrow -1$.
  • ...and 1 more figures

Theorems & Definitions (38)

  • Definition 1
  • Proposition 1: See, for example, boscain-rossi, Prop. 2
  • Remark 1
  • Definition 2
  • Remark 2
  • Proposition 2
  • proof
  • Definition 3
  • Proposition 3: podobryaev-sachkov-so3, formula (4)
  • Proposition 4: bates-fasso
  • ...and 28 more