Hyperbolic isometry

In mathematics , hyperbolic isometrics 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 a hyperbolic isometry if it does not have a fixed point , but there is one under invariant geodesic . ${\ displaystyle f}$

In particular, a hyperbolic isometry has two fixed points at infinity .

example

Let be the half-plane model of the hyperbolic plane and one 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) = \ lambda z}

with given figure. One can check that it is an isometry and that the geodesics are through and invariant. So it's a hyperbolic isometry. ${\ displaystyle \ lambda> 1}$ ${\ displaystyle f}$ ${\ displaystyle 0}$ ${\ displaystyle \ infty}$

More generally, isometries of the hyperbolic plane can be described by matrices and isometries of the 3-dimensional hyperbolic space by matrices . In the case of the hyperbolic plane, the isometry described by a matrix is hyperbolic 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 \ mid \ operatorname {Sp} (A) \ mid> 2}

applies. In this case , this condition is sufficient but not necessary for hyperbolic isometry. The above example corresponds to the matrix . ${\ displaystyle A \ in SL (2, \ mathbb {C})}$ ${\ displaystyle \ left ({\ begin {matrix} {\ sqrt {\ lambda}} & 0 \\ 0 & {\ frac {1} {\ sqrt {\ lambda}}} \ end {matrix}} \ right)}$

Equivalent characterization

For an isometry, let it be defined by ${\ displaystyle f \ colon X \ to X}$ ${\ displaystyle d_ {f} \ colon X \ to \ mathbb {R} _ {\ geq 0}}$

{\ displaystyle d_ {f} (x) = d (f (x), x)}

.

Isometry is hyperbolic if and only if there is a with ${\ displaystyle x_ {0} \ in X}$

{\ displaystyle d_ {f} (x_ {0}) = \ inf \ left \ {d_ {f} (x) \ colon x \ in X \ right \}}

and this infimum is positive .

The amount

{\ displaystyle \ operatorname {Min} (f) = \ left \ {z \ in X \ colon d_ {f} (z) = \ inf \ left \ {d_ {f} (x) \ colon x \ in X \ right \} \ right \} \ subset X}

is then a union of invariant geodesics.

Loxodromic isometrics

If the hyperbolic space is with , then the hyperbolic isometries defined above are also referred to as loxodromic isometries . Hyperbolic isometries are only those loxodromic isometries that act as transvections along an invariant geodesic, i.e. do not cause any rotation around these geodesics. ${\ displaystyle X = {\ mathbb {H}} ^ {n}}$ ${\ displaystyle n \ geq 3}$

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