The Spheres of Sol

TL;DR

This paper provides a detailed geometric analysis of the solvable Lie group Sol, describing its cut locus, exponential map domain, and the topology and singularities of its metric spheres.

Contribution

It offers the first precise description of the cut locus and exponential map domain for Sol, and characterizes the topology and singular points of its metric spheres.

Findings

Metric spheres in Sol are topological spheres.

The cut locus of the identity in Sol is explicitly described.

Singular points of the spheres are characterized almost exactly.

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.