The spheres of Sol

Coiculescu, Matei P¹ ; Schwartz, Richard Evan²

¹ Department of Mathematics, Princeton University, Princeton, NJ, United States
² Department of Mathematics, Brown University, Providence, RI, United States

Geometry & topology, Tome 26 (2022) no. 5, pp. 2103-2134.

Résumé

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

Let $E:{ℝ}^3\to \mathrm{Sol}$ be the Riemannian exponential map. Given $V=(x,y,z)\in {ℝ}^3$, let ${\gamma }_V=\{E(tV)\mid t\in [0,1]\}$ 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

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.

Détail

DOI : 10.2140/gt.2022.26.2103

Keywords: Sol, spheres, geodesics, cut locus

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