The Spheres of Sol
Differential Geometry
2022-12-21 v8 Geometric Topology
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.
Cite
@article{arxiv.1911.04003,
title = {The Spheres of Sol},
author = {Matei P. Coiculescu and Richard Evan Schwartz},
journal= {arXiv preprint arXiv:1911.04003},
year = {2022}
}
Comments
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