论文标题
加权Fermat-Frechet的等距嵌入多核能用于$ K $ -Space中的单纯式边界到Frechet MultiSimplex的等级变形
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
论文作者
论文摘要
在本文中，我们研究了$ \ frac {n（n+1）} {2} {2} - $正实数的$ tuple $ n $ n $ simplexes $ n $ n $ k $ k $ -space（$ n $ n $ dimensional-dimensional euclidean euclidean space $ \ n $ kn） $ n $ -dimensional radius $ \ frac {1} {\ sqrt {k}} $（$ \ mathbb {s} _ {\ frac {\ frac {1} {\ sqrt $ \ mathbb {h} _ {k}^{n} $ of constance curvature $ k $如果$ k <0 $）。 （加权的）Fermat-Frechet问题是对$ n $ simplexes的（加权）Fermat问题的新概括。我们通过使用某些条件对Dekster-Wilker发现的边缘长度使用一些条件来控制加权的Fermat-Frechet问题的解决方案（加权费马特树）的数量。为了在$ k $ - 空间中构建一个加权费马特 - 弗雷切特多的沉浸式，我们使用$ n $ n $ simplexes的godel-schoenberg的等速线沉浸在$ n $ sphere中的$ n $ simplexes，gromov和gromov insmetric insmetric and gromov and gromov fors for Adgsift fors for for aftive fors for for for for for for for fersy fermat fermat fermat in $ n $ n $ n $ n $ n $ n $ n $ n $ n $ n $ n $ n $ n $ n $ n $ n $ n $ $ \ mathbb {h} _ {k}^{n} $。最后，我们创建了一种新的变分方法，该方法与Schafli，Luo和Milnor的技术不同，以在3 $ K $ -SPACE中相对于可变的大地测量弧而区分地球弧的长度。通过应用此方法，我们从一个方程系统中消除了一个可变的测量弧，该方程式为加权的Fermat-Frechet溶液提供了边缘长度确定（FRECHET）四面体的分解。
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.
