Simplexes in Riemannian manifolds

Abstract:

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 ${p_0},{p_1}, \ldots ,{p_m}$ in ${M^n},m \leqslant n$, 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 ${R^n}:{p_0}$ is a prescribed point, the shortest path ${p_0}{p_1}$ has a prescribed direction at ${p_0}$, the triangle ${p_0}{p_1}{p_2}$ determines a prescribed $2$-dimensional direction at ${p_0}$, and so on.