Skip to main content Accessibility help
×
Hostname: page-component-78c5997874-94fs2 Total loading time: 0 Render date: 2024-11-09T20:04:55.588Z Has data issue: false hasContentIssue false

6 - Isometry groups

Published online by Cambridge University Press:  05 June 2015

Anthony G. O'Farrell
Affiliation:
National University of Ireland, Maynooth
Ian Short
Affiliation:
The Open University, Milton Keynes
Get access

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
Publisher: Cambridge University Press
Print publication year: 2015

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Save book to Kindle

To save this book to your Kindle, first ensure [email protected] is added to your Approved Personal Document E-mail List under your Personal Document Settings on the Manage Your Content and Devices page of your Amazon account. Then enter the ‘name’ part of your Kindle email address below. Find out more about saving to your Kindle.

Note you can select to save to either the @free.kindle.com or @kindle.com variations. ‘@free.kindle.com’ emails are free but can only be saved to your device when it is connected to wi-fi. ‘@kindle.com’ emails can be delivered even when you are not connected to wi-fi, but note that service fees apply.

Find out more about the Kindle Personal Document Service.

  • Isometry groups
  • Anthony G. O'Farrell, National University of Ireland, Maynooth, Ian Short, The Open University, Milton Keynes
  • Book: Reversibility in Dynamics and Group Theory
  • Online publication: 05 June 2015
  • Chapter DOI: https://doi.org/10.1017/CBO9781139998321.007
Available formats
×

Save book to Dropbox

To save content items to your account, please confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your account. Find out more about saving content to Dropbox.

  • Isometry groups
  • Anthony G. O'Farrell, National University of Ireland, Maynooth, Ian Short, The Open University, Milton Keynes
  • Book: Reversibility in Dynamics and Group Theory
  • Online publication: 05 June 2015
  • Chapter DOI: https://doi.org/10.1017/CBO9781139998321.007
Available formats
×

Save book to Google Drive

To save content items to your account, please confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your account. Find out more about saving content to Google Drive.

  • Isometry groups
  • Anthony G. O'Farrell, National University of Ireland, Maynooth, Ian Short, The Open University, Milton Keynes
  • Book: Reversibility in Dynamics and Group Theory
  • Online publication: 05 June 2015
  • Chapter DOI: https://doi.org/10.1017/CBO9781139998321.007
Available formats
×