## $$m$$-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.

### MSC:

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