  In this paper, we study the weighted Fermat-Frechet problem for a $\frac{N
(N+1)}{2}-$tuple of positive real numbers determining $N$-simplexes in the $N$
dimensional $K$-Space ($N$-dimensional Euclidean space $\mathbb{R}^{N}$ if
$K=0,$ the $N$-dimensional open hemisphere of radius $\frac{1}{\sqrt{K}}$
($\mathbb{S}_{\frac{1}{\sqrt{K}}}^{N}$) if $K >0$ and the Lobachevsky space
$\mathbb{H}_{K}^{N}$ of constant curvature $K$ if $K<0$). The (weighted)
Fermat-Frechet problem is a new generalization of the (weighted) Fermat problem
for $N$-simplexes. We 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 $\mathbb{H}_{K}^{N}$. 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 3$K$-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.
