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The spheres of Sol

### Matei P Coiculescu and Richard Evan Schwartz

Geometry & Topology 26 (2022) 2103–2134
##### Abstract

Let Sol be the $3$–dimensional solvable Lie group whose underlying space is ${ℝ}^{3}$ and whose left-invariant Riemannian metric is given by

 ${e}^{-2z}d{x}^{2}+{e}^{2z}d{y}^{2}+d{z}^{2}.$

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

 $\mathrm{AGM}\left(\sqrt{|xy|},\frac{1}{2}\sqrt{{\left(|x|+|y|\right)}^{2}+{z}^{2}}\right)\le \pi .$

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
Primary: 53C30
##### 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