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The spheres of Sol

Matei P Coiculescu and Richard Evan Schwartz

Geometry & Topology 26 (2022) 2103–2134

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

e2z dx2 + e2z dy2 + dz2.

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

AGM (|xy|, 1 2(|x| + |y|)2 + z2) 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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Sol, spheres, geodesics, cut locus
Mathematical Subject Classification
Primary: 53C30
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
Matei P Coiculescu
Department of Mathematics
Princeton University
Princeton, NJ
United States
Richard Evan Schwartz
Department of Mathematics
Brown University
Providence, RI
United States