Isometric embedding of a weighted Fermat-Frechet multitree for isoperimetric deformations of the boundary of a simplex to a Frechet multisimplex in the $K$-Space

Abstract

In this paper, we study the weighted Fermat-Frechet problem for a N (N +1) 2 −tuple of positive real numbers determining N -simplexes in theWe control the number of solutions (weighted Fermat trees) with respect to the weighted Fermat-Frechet problem that we call a weighted Fermat-Frechet multitree, by using some conditions for the edge lengths discovered by Dekster-Wilker. In order to construct an isometric immersion of a weighted Fermat-Frechet multitree in the K-Space, we use the isometric immersion of Godel-Schoenberg for N -simplexes in the N -sphere and the isometric immersion of Gromov (up to an additive constant) for weighted Fermat (Steiner) trees in the N -hyperbolic space H N K . Finally, we create a new variational method, which differs from Schafli's, Luo's and Milnor's techniques to differentiate the length of a geodesic arc with respect to a variable geodesic arc, in the 3K-Space. By applying this method, we eliminate one variable geodesic arc from a system of equations, which give the weighted Fermat-Frechet solution for a sextuple of edge lengths determining (Frechet) tetrahedra.