The Spheres of Sol
arXiv:1911.04003 · doi:10.2140/gt.2022.26.2103
Abstract
Let Sol be the three-dimensional solvable Lie group equipped with its standard left-invariant Riemannian metric. We give a precise description of the cut locus of the identity, and a maximal domain in the Lie algebra on which the Riemannian exponential map is a diffeomorphism. As a consequence, we prove that the metric spheres in Sol are topological spheres, and we characterize their singular points almost exactly.
36 pages, 6 figures. This version is a revision. We revised it according to the referee reports from our Geometry&Topology submission of the first version. The exposition is somewhat cleaner, especially in Chapter 3, and we have added some new diagrams and computer plots