Elliptical isometry
In mathematics , elliptical isometries are important in hyperbolic geometry and, more generally, in the theory of CAT (0) spaces .
definition
Let it be a complete CAT (0) space , for example a hyperbolic space . An isometry
is an elliptical isometry if it has a fixed point , i. H. if there is one with .
example
Let be the half-plane model of the hyperbolic plane and the through
given figure. You can check that there is an isometric drawing and that it has the fixed point . So it's an elliptical isometry.
More generally, isometries of the hyperbolic plane can be described by matrices and isometries of the 3-dimensional hyperbolic space by matrices . An isometry of the hyperbolic plane described by is elliptical if and only if the inequality for the trace of
applies. For an elliptical isometry of the hyperbolic space described by it necessarily applies
- and .
properties
Let it be a complete CAT (0) space and an isometry .
- is elliptical if and only if it has a bounded orbit .
- is elliptical if and only if there is one for which is elliptical.
See also
literature
- Martin Bridson , André Haefliger : Metric spaces of non-positive curvature. Basic teaching of the mathematical sciences 319. Springer-Verlag, Berlin 1999, ISBN 3-540-64324-9 .
- Francis Bonahon : Low-dimensional geometry. From Euclidean surfaces to hyperbolic knots. Student Mathematical Library, 49th IAS / Park City Mathematical Subseries. American Mathematical Society, Providence, RI; Institute for Advanced Study (IAS), Princeton, NJ, 2009. ISBN 978-0-8218-4816-6