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.

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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