Beiträge zur Algebra und Geometrie / Contributions to Algebra and GeometryVol. 40, No. 1, pp. 261-266 (1999)

$m$-Point Invariants of Real Geometries

Roland Höfer

Mathematisches Seminar der Universität Hamburg, Bundesstrasse 55, D-20146 Hamburg, Germany

Abstract: Let $V$ be a finite dimensional vector space over a field $K$ with a non-degenerate symmetric, alternating, or hermitian scalar product. We characterize the orbits of $m$-tuples of vectors of $V$ under the group of isometries of $V$ by the Gram matrix and an additional subspace of $K^m$. This characterization will then be applied to determine all $m$-point invariants of real euclidean, spherical, hyperbolic, and de-Sitter geometries.

Classification (MSC91): 15A63, 51M10, 83A05

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