The spheres of Sol

Let $E : ℝ^{3} \to Sol$ be the Riemannian exponential map. Given $V = (x, y, z) \in ℝ^{3}$ , let $γ_{V} = {E (t V) ∣ t \in [0, 1]}$ be the corresponding geodesic segment. Let AGM stand for the arithmetic–geometric mean. We prove that $γ_{V}$ is a distance-minimizing segment in Sol if and only if

AGM (\sqrt{| x y |}, \frac{1}{2} \sqrt{{(| x | + | y |)}^{2} + z^{2}}) \leq π .

We use this inequality to precisely characterize the cut locus in Sol, prove that the metric spheres in Sol are topological spheres, and almost exactly characterize their singular sets.

Keywords

Sol, spheres, geodesics, cut locus

Mathematical Subject Classification

Primary: 53C30

References

Publication

Received: 6 August 2020

Revised: 25 April 2021

Accepted: 21 June 2021

Published: 12 December 2022

Proposed: David M Fisher

Seconded: Dmitri Burago, David Gabai

Authors

Matei P Coiculescu
Department of Mathematics Princeton University Princeton, NJ United States

Richard Evan Schwartz
Department of Mathematics Brown University Providence, RI United States

Volume 26, issue 5 (2022)

Matei P Coiculescu and Richard Evan Schwartz

Abstract

Keywords

Mathematical Subject Classification

References

Publication

Authors