Geom. Topol. 26 (5), 2103-2134, (2022) DOI: 10.2140/gt.2022.26.2103
Matei P Coiculescu, Richard Evan Schwartz
KEYWORDS: Sol, spheres, geodesics, Cut locus, 53C30
Let Sol be the –dimensional solvable Lie group whose underlying space is and whose left-invariant Riemannian metric is given by
Let be the Riemannian exponential map. Given , let be the corresponding geodesic segment. Let AGM stand for the arithmetic–geometric mean. We prove that 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.