Simplexes in spaces of constant curvature

Abstract

In hyperbolic, Euclidean and spherical n-space, we determine, for each positive number l, the largest interval of the form ),.(/) ~< l u ~< l which guarantees the existence of an nsimplex Pl P2 "'" P,+t with edge-lengths pipj=lu. (In spherical geometry of curvature 1 the interval is empty unless 1 ~< 2 arcsin ~.)The assertion that these intervals are as large as possible is justified because each of them allows certain degenerate simplexes. We determine explicitly all of these critical configurations.

The Jung Theorem for spherical and hyperbolic spaces

“…The equality in estimate (1.3) actually appears for S = ao. This follows from Theorem 2 in [5]. One has thus…”

Simplexes in Riemannian Manifolds

“…A simple sufficient condition of this existence established there is, roughly speaking, that the edge lengths do not differ too much, see [3,Theorem 2]. We deal here with m + 1 points Po, P\, ■■■ , Pm in a Riemannian n-manifold M" , m < n, with prescribed mutual distances fj and establish a condition on the matrix (//,) under which the points p, can be selected as freely as in R" : po is a prescribed point, the shortest path poP\ has a prescribed direction at po , the triangle PoP\P2 determines a prescribed 2-dimensional direction at po , and so on.…”

“…Existence of a simplex with prescribed edge lengths in Euclidean, spherical, and hyperbolic spaces was studied in [3]. A simple sufficient condition of this existence established there is, roughly speaking, that the edge lengths do not differ too much, see [3,Theorem 2].…”

“…where ( 2/m if wis even, (We use m -1 as the integer argument to comply with [3].) This I has the property that each allowable m x m matrix (a,y) with \ho hx 0/ V1-58 3-15 0 / where 1.58 > n/2 and 3.15 > n. This matrix has realizations in M3, say, those located in 7?2.…”

Simplexes in Riemannian manifolds

“…Existence of a simplex with prescribed edge lengths in Euclidean, spherical, and hyperbolic spaces was studied in [3]. A simple sufficient condition of this existence established there is, roughly speaking, that the edge lengths do not differ too much, see [3,Theorem 2]. We deal here with m + 1 points Po, P\, ■■■ , Pm in a Riemannian n-manifold M" , m < n, with prescribed mutual distances fj and establish a condition on the matrix (//,) under which the points p, can be selected as freely as in R" : po is a prescribed point, the shortest path poP\ has a prescribed direction at po , the triangle PoP\P2 determines a prescribed 2dimensional direction at po , and so on.…”