A geometric construction of the K-loop of a hyperbolic space |
| |
Authors: | Helmut Karzel Heinrich Wefelscheid |
| |
Institution: | (1) Mathematisches Institut, Technische Universität München, D-80290 Munich, Germany;(2) Fachbereich 11: Mathematik, Universität Duisburg, D-47048 Duisburg, Germany |
| |
Abstract: | It is well known that the homogeneous orthochronous proper Lorentzgroup is isomorphic to the proper motion group of the hyperbolic space. To each Lorentz boost \ {id} there corresponds in the hyperbolic space exactly one lineL
such that fixes each of the two ends ofL
. Furthermore has no fixed points but each plane containingL
is fixed by . If we fix a pointo, then to each other pointa there is exactly one boosta
+ such thatL
a+ is the line joiningo anda anda
+(o)=a. The set P of points of the hyperbolic space is turned in a K-loop (P, +) bya+b:=a
+(b). Each line of the hyperbolic space has the representationa+Z(b) wherea, b P,b 0 andZ(b):= {x P |x+b=b+x}.Dedicated to H. Salzmann on the occasion of his 65th birthdaySupported by the NATO Scientific Affairs Division grant CRG 900103. |
| |
Keywords: | 51A25 20N05 |
本文献已被 SpringerLink 等数据库收录! |
|