Abstract
In this paper we give a definition of a Möbius-space (i.e. a system of “points” and “circles”) and establish a higher-dimensional circle-geometry analogous to the synthetic construction of projective geometry. For each point of a Möbius-space there exists an affine substructure, the dimension of which is equal to that of the Möbius-space. A Möbius-space of dimension ≥3 is egglike, hence its spheres satisfy the Bundle Theorem [12], p. 758.
In our definition we exclude those minimal models of geometries, which correspond in their algebraic description to the prime field of characteristic 2.
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Literatur
BAER, R.: Linear algebra and projective geometry. New York: Acad. Press 1952.
BENZ, W.: Über Möbiusebenen. Ein Bericht. Jber. dtsch. Math.-Ver.63, 1–27 (1960).
DEMBOWSKI, P.: Finite geometries. Berlin-Heidelberg-New York: Springer 1968.
HOFFMAN, A.J.: On the foundations of inversion geometry. Trans.Amer.Math.Soc.71, 218–242 (1951).
IZUMI, S.: Lattice theoretical foundation of circle geometry. Proc.Imp.Acad.Tokyo16, 515–517 (1940).
KARZEL, H.: Inzidenzgruppen II. Ausarbeitung Hamburg 1966.
KARZEL, H.: Grundlagen der Geometrie. Ausarbeitung Karlsruhe 1968.
MÄURER, H.: Möbius- und Laguerre-Geometrien über schwach konvexen Semiflächen. Math.Z.98, 355–386 (1967).
MÄURER, H.: Ein axiomatischer Aufbau der mindestens 3-dimensionalen Möbius-Geometrie. Math.Z.103,282–305 (1968).
PIERI, M.: Nuovi principii di geometria delle inversioni. Giornale di Mathematiche di Battaglini49, 49–96 (1911) und50, 106–140 (1912).
TITS, J.: Ovoides et groupes de Suzuki. Arch.Math.13, 187–198 (1962).
WAERDEN, B.L. van der u. L.J. SMID: Eine Axiomatik der Kreisgeometrie und der Laguerregeometrie. Math. Ann.110, 753–776 (1935).
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Werner, H. Eine Definition des Möbiusraumes. Manuscripta Math 2, 39–47 (1970). https://doi.org/10.1007/BF01168478
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DOI: https://doi.org/10.1007/BF01168478