\(m\)-point invariants of real geometries. (English) Zbl 0959.15032

Let \(K\) be a skewfield with \(\text{ char} K\neq 2\) and let \(V\) be a finite-dimensional vector space over \(K.\) Let \((\cdot,\cdot)\) be a sesquilinear form (symmetric, alternating, or Hermitian). Let \((x_1,\ldots,x_m)\) be an \(m\)-tuple of vectors in \(V.\) If \(\sigma\) is an isometry of \(V,\) put \(\sigma x_i={x_i}'.\) The author is interested in the orbits of \(m\)-tuples under the group of isometries of \(V.\)
Clearly, a necessary condition for an \(m\)-tuple to be in this orbit is that the Gram matrices of \((x_1,\ldots,x_m)\) and \(({x_1}',\ldots,{x_m}')\) are equal, i.e., \((x_i,x_j)=({x_i}',{x_j}').\)
The author obtains necessary and sufficient conditions for \(({x_1}',\ldots,{x_m}')\) to be in the orbit of \((x_1,\ldots,x_m),\) provided \(K\) is an ordered commutative field whose positive elements are squares. He applies this result to determine all \(m\)-point invariants of real euclidean, spherical, hyperbolic, and de Sitter geometries.


15A63 Quadratic and bilinear forms, inner products
83A05 Special relativity
51M10 Hyperbolic and elliptic geometries (general) and generalizations
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