$K$ -空间中单形到Frechet多重单形边界等周变形的加权Fermat-Frechet多重树的等距嵌入
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
摘要：
本文研究了$N$维$K$ -空间（$N$维欧氏空间$\mathbb{R}^{N}$，如果$K=0,$，$N$维开半球半径$\frac{1}{\sqrt{K}}$（$\mathbb{S}_{\frac{1}{\sqrt{K}}}^{N}$），如果$K >0$和常曲率的罗巴切夫斯基空间$\mathbb{H}_{K}^{N}$，如果$K<0$）中确定$N$ -单形的正实数$\frac{N (N+1)}{2}-$元组的加权Fermat-Frechet问题.（加权）Fermat-Frechet问题是$N$ -单形的（加权）Fermat问题的一个新的推广。我们控制的解决方案（加权费马树）的数量相对于加权费马-弗雷歇问题，我们称之为加权费马-弗雷歇多树，通过使用一些条件的边长发现Dekster-Wilker。为了构造$K$ -空间中加权Fermat-Frechet多重树的等距浸入，我们利用了$N$ -球面中$N$ -单形的Godel-Schoenberg的等距浸入和$N$ -双曲空间$\mathbb{H}_{K}^{N}$中加权Fermat（Steiner）树的Gromov的等距浸入（直到加常数）.最后，我们创建了一个新的变分方法，它不同于Schafli的，罗的和Milnor的技术，以区分长度的测地弧相对于可变测地弧，在3 $K$ -空间。应用这种方法，我们从方程组中消去了一个变量测地弧，从而给出了一个六重边长决定（Frechet）四面体的加权Fermat-Frechet解。
英文摘要：
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.
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