SIMPLEXES IN RIEMANNIAN-MANIFOLDS

Existence of a simplex with prescribed edge lengths in Euclidean, spherical, and hyperbolic spaces was studied recently. A simple sufficient condition of this existence is, roughly speaking, that the lengths do not differ too much. We extend these results to Riemannian n-manifolds M(n). More precisely we consider m + 1 points p0, p1, ... , p(m) in M(n), m less-than-or-equal-to n, with prescribed mutual distances l(ij) and establish a condition on the matrix (l(ij)) under which the points p(i) can be selected as freely as in R(n) : p0 is a prescribed point, the shortest path p0p1 has a prescribed direction at p0, the triangle p0p1p2 determines a prescribed 2-dimensional direction at p0, and so on.