6 - Isometry groups

Summary

Isometries of spherical, Euclidean, and hyperbolic space

For each positive integer n, there are three simply-connected, complete Rie-mannian n-manifolds with constant curvature, namely n-dimensional spherical space Sⁿ, n-dimensional Euclidean space ℝⁿ, and n-dimensional hyperbolic space ℍⁿ. Note that ℝ¹ and ℍ¹ are isometric, but otherwise there are no repetitions in this list. We denote the isometry groups of these manifolds by Isom(Sⁿ), Isom(ℝⁿ), and Isom(ℍⁿ). These groups are each generated by reflections. We denote the three subgroups of these three isometry groups, comprised of orientation-preserving isometries, by Isom⁺ (Sⁿ), Isom⁺ (ℝRⁿ), and Isom⁺ (ℍⁿ). A map in Isom(Sⁿ) lies in Isom⁺ (Sⁿ) if and only if it can be expressed as a composite of an even number of reflections. Similar comments apply to the groups Isom⁺ (ℝⁿ) and Isom⁺ (ℍⁿ).

We studied the orthogonal group O(n, ℝ) and the special orthogonal group SO(n, ℝ) in Chapter 4; these two groups are Isom(Sⁿ⁻¹) and Isom⁺ (Sⁿ⁻¹), respectively. In this chapter we consider reversibility in the remaining four isometry groups Isom(ℝⁿ), Isom(ℍⁿ), Isom⁺ (ℝⁿ), and Isom⁺ (ℍⁿ).

Hyperbolic geometry in two and three dimensions

In Chapter 1 we briefly discussed reversibility in the Euclidean isometry groups Isom⁺ (ℝ²) and Isom⁺ (ℝ³). We found that, in two dimensions, the only elements that are strongly reversible, other than involutions, are translations. In three dimensions we found that all isometries are strongly reversible. Before we tackle higher-dimensional isometry groups we first, in this section, consider isometry groups of two- and three-dimensional hyperbolic space.