A characterization of dimension-free hyperbolic geometry and the functional equation of 2-point invariants

Autor: Walter Benz
Rok vydání: 2010
Předmět:
Zdroj: Aequationes mathematicae. 80:5-11
ISSN: 1420-8903
0001-9054
DOI: 10.1007/s00010-010-0050-1
Popis: Let X be a real inner product space of arbitrary (finite or infinite) dimension ≥ 2. Define $$P:=\{x\in X\mid\lVert x\rVert < 1\}\qquad\qquad(1)$$ and G to be the group of all bijections of P such that the images and pre-images of the following sets, called P-lines, $$(a,b):=\left\{x\in X\backslash\{a,b\} \mid\lVert a-b\rVert=\lVert a-x\rVert+\lVert x-b\rVert\right\},\qquad\qquad(2)$$ \({a,b\in X,\,a\neq b,\| a\|=1=\|b\|}\), are again P-lines. Observe that (a, b) is given by the segment $$(a,b)=\{a +\varrho(b-a)\mid 0 < \varrho < 1\},$$ where the two points a ≠ b are on the ball B(0, 1). In Theorem 1 we prove that the geometry (P, G) is isomorphic to the hyperbolic geometry (X, M(X, hyp)) over X (see Sect. 1). In Theorem 2 we solve the Functional Equation of 2-point invariants for (P, G), showing that the notion of hyperbolic distance for (P, G) stemming from the isomorphism of Theorem 1 must be a basis of all its 2-point invariants. For definitions see the book [Benz in Classical Geometries in Modern Contexts. Geometry of Real Inner Product Spaces. Birkhauser, Basel, 1st edn (2005) (2nd edn, 2007)], especially Sect. 2.12 and 5.11.
Databáze: OpenAIRE
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