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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 problemfor $N$-simplexes. We control the number of solutions (weighted Fermat trees)with respect to the weighted Fermat-Frechet problem that we call a weightedFermat-Frechet multitree, by using some conditions for the edge lengthsdiscovered by Dekster-Wilker. In order to construct an isometric immersion of aweighted Fermat-Frechet multitree in the $K$- Space, we use the isometricimmersion of Godel-Schoenberg for $N$-simplexes in the $N$-sphere and theisometric immersion of Gromov (up to an additive constant) for weighted Fermat(Steiner) trees in the $N$-hyperbolic space $\mathbb{H}_{K}^{N}$. Finally, wecreate a new variational method, which differs from Schafli's, Luo's andMilnor's techniques to differentiate the length of a geodesic arc with respectto a variable geodesic arc, in the 3$K$-Space. By applying this method, weeliminate one variable geodesic arc from a system of equations, which give theweighted Fermat-Frechet solution for a sextuple of edge lengths determining(Frechet) tetrahedra.

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