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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 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.

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