Hyperbolic spaces and quadratic forms

even investigated spaces with distances not necessarily real numbers. Menger [5], [6] investigated necessary and sufficient conditions that a semimetric (n+ 1)-tuple be isometrically imbeddable in a euclidean n-dimensional space En. He solved this problem by means of equations and inequalities involving certain determinants. Blumenthal and Garrett [3] and Blumenthal [2] solved the similar problems for n-dimensional spherical space S, of radius r and n-dimensional hyperbolic space Hn,r of curvature -1/r2, r>0 by means of certain determinants. Schoenberg [7], [8] gave complete and independent solutions for the problems for euclidean and spherical spaces by means of quadratic forms. The purpose of this paper is to solve the problem for hyperbolic space by means of quadratic forms. A related problem in hyperbolic space is solved. The solution to this problem is quite trivial in the setting of quadratic forms, while it would be quite difficult if only the determinant conditions of Blumenthal were available. 2. The criterion. A complete solution to the problem of necessary and sufficient conditions that a semimetric (n+1)-tuple be isometrically imbeddable in hyperbolic space is given by the following theorem. THEOREM. A necessary and sufficient condition that a semimetric (n+ 1)-tuple, Po, Pi, , p. n be congruently imbeddable in Hk,, and not in Hk-l,t is that the quadratic form