Book contents
- Frontmatter
- Contents
- Preface
- 1 Origins
- 2 Basic ideas
- 3 Finite groups
- 4 The classical groups
- 5 Compact groups
- 6 Isometry groups
- 7 Groups of integer matrices
- 8 Real homeomorphisms
- 9 Circle homeomorphisms
- 10 Formal power series
- 11 Real diffeomorphisms
- 12 Biholomorphic germs
- References
- List of frequently used symbols
- Index of names
- Subject index
6 - Isometry groups
Published online by Cambridge University Press: 05 June 2015
- Frontmatter
- Contents
- Preface
- 1 Origins
- 2 Basic ideas
- 3 Finite groups
- 4 The classical groups
- 5 Compact groups
- 6 Isometry groups
- 7 Groups of integer matrices
- 8 Real homeomorphisms
- 9 Circle homeomorphisms
- 10 Formal power series
- 11 Real diffeomorphisms
- 12 Biholomorphic germs
- References
- List of frequently used symbols
- Index of names
- Subject index
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.
- Type
- Chapter
- Information
- Reversibility in Dynamics and Group Theory , pp. 92 - 104Publisher: Cambridge University PressPrint publication year: 2015