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

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Erscheinungsjahr:
2010
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Text
Schlagworte:
  • Real inner product spaces
  • dimension-free
  • functional equation of 2-point invariants
  • hyperbolic geometry
  • isomorphic geometries
Beschreibung:
  • Let X be a real inner product space of arbitrary (finite or infinite) dimension ≥ 2. Define 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, are again P-lines. Observe that (a, b) is given by the segment 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. Birkhäuser, Basel, 1st edn (2005) (2nd edn, 2007)], especially Sect. 2. 12 and 5. 11. © 2010 Springer Basel AG.
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