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 ${\displaystyle X}$

${\displaystyle f\colon X\to X}$

is an elliptical isometry if it has a fixed point , i. H. if there is one with . ${\displaystyle x\in X}$ ${\displaystyle f(x)=x}$

example

Let be the half-plane model of the hyperbolic plane and the through ${\displaystyle X={\mathbf {H} }^{2}=\left\{z\in \mathbb {C} \colon \operatorname {Im} (z)>0\right\}}$ ${\displaystyle f\colon X\to X}$

${\displaystyle f(z)={\frac {\cos(\phi )z+\sin(\phi )}{-\sin(\phi )z+\cos(\phi )}}}$

given figure. You can check that there is an isometric drawing and that it has the fixed point . So it's an elliptical isometry. ${\displaystyle f}$ ${\displaystyle z=i}$

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 ${\displaystyle A\in SL(2,\mathbb {R} )}$ ${\displaystyle A\in SL(2,\mathbb {C} )}$ ${\displaystyle A\in SL(2,\mathbb {R} )}$ ${\displaystyle A}$

${\displaystyle -2<\operatorname {Sp} (A)<2}$

applies. For an elliptical isometry of the hyperbolic space described by it necessarily applies ${\displaystyle A\in SL(2,\mathbb {C} )}$

${\displaystyle \mid \operatorname {Sp} (A)\mid \leq 2}$and . ${\displaystyle \operatorname {Sp} (A)\not \in \left\{-2,2\right\}}$

properties

Let it be a complete CAT (0) space and an isometry . ${\displaystyle X}$ ${\displaystyle f\colon X\to X}$

• ${\displaystyle f}$is elliptical if and only if it has a bounded orbit .
• ${\displaystyle f}$is elliptical if and only if there is one for which is elliptical. ${\displaystyle n>0}$ ${\displaystyle f^{n}}$

See also

• Hyperbolic isometry
• Parabolic isometry

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

Web links

• Chang: Isometries of the hyperbolic plane

Individual evidence

1. ↑ Bridson-Haefliger, op.cit., Proposition 6.7
2. ↑ ibd.