In this paper, we study the weighted Fermat-Frechet problem for a <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$\frac{N (N+1)}{2}$</EquationSource> <EquationSource Format="MATHML"><math> <mfrac> <mrow> <mi>N</mi> <mo stretchy="false">(</mo> <mi>N</mi> <mo>+</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mrow> <mn>2</mn> </mfrac> </math></EquationSource> </InlineEquation>-tuple of positive real numbers determining N-simplexes or an N-simplex in the N-dimensional K-space (N-dimensional Euclidean space <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$\mathbb {R}^{N}$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mrow> <mi mathvariant="double-struck">R</mi> </mrow> <mi>N</mi> </msup> </math></EquationSource> </InlineEquation> if <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$K=0$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>K</mi> <mo>=</mo> <mn>0</mn> </mrow> </math></EquationSource> </InlineEquation>, the N-dimensional open hemisphere of radius <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$\frac{1}{\sqrt{K}}$</EquationSource> <EquationSource Format="MATHML"><math> <mfrac> <mn>1</mn> <msqrt> <mi>K</mi> </msqrt> </mfrac> </math></EquationSource> </InlineEquation> (<InlineEquation ID="IEq5"> <EquationSource Format="TEX">$\mathbb {S}_{\frac{1}{\sqrt{K}}}^{N}$</EquationSource> <EquationSource Format="MATHML"><math> <msubsup> <mi mathvariant="double-struck">S</mi> <mrow> <mfrac> <mn>1</mn> <msqrt> <mi>K</mi> </msqrt> </mfrac> </mrow> <mi>N</mi> </msubsup> </math></EquationSource> </InlineEquation>) if <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$K >0$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>K</mi> <mo>></mo> <mn>0</mn> </mrow> </math></EquationSource> </InlineEquation> and the Lobachevsky space <InlineEquation ID="IEq7"> <EquationSource Format="TEX">$\mathbb {H}_{K}^{N}$</EquationSource> <EquationSource Format="MATHML"><math> <msubsup> <mi mathvariant="double-struck">H</mi> <mrow> <mi>K</mi> </mrow> <mi>N</mi> </msubsup> </math></EquationSource> </InlineEquation> of constant curvature K if <InlineEquation ID="IEq8"> <EquationSource Format="TEX">$K<0$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>K</mi> <mo><</mo> <mn>0</mn> </mrow> </math></EquationSource> </InlineEquation>). 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. 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 <InlineEquation ID="IEq9"> <EquationSource Format="TEX">$\mathbb {H}_{K}^{N},$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msubsup> <mi mathvariant="double-struck">H</mi> <mrow> <mi>K</mi> </mrow> <mi>N</mi> </msubsup> <mo>,</mo> </mrow> </math></EquationSource> </InlineEquation> in order to construct an isometric immersion of a weighted Fermat-Frechet multitree in the K-space. Finally, we create a new variational method, which differs from Schlafli’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 gives the weighted Fermat-Frechet solution for a sextuple of edge lengths determining (Frechet) tetrahedra.

Isometric embeddings of Fermat-Frechet “multitrees" in the K-space

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发表日期：

2026年5月15日

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Anastasios N. Zachos

摘要

In this paper, we study the weighted Fermat-Frechet problem for a $\frac{N (N+1)}{2}$ $\frac{N (N + 1)}{2}$ -tuple of positive real numbers determining N-simplexes or an N-simplex in the N-dimensional K-space (N-dimensional Euclidean space $\mathbb {R}^{N}$ $R^{N}$ if $K=0$ $K = 0$ , the N-dimensional open hemisphere of radius $\frac{1}{\sqrt{K}}$ $\frac{1}{\sqrt{K}}$ ( $\mathbb {S}_{\frac{1}{\sqrt{K}}}^{N}$ $S_{\frac{1}{\sqrt{K}}}^{N}$ ) if $K >0$ $K > 0$ and the Lobachevsky space $\mathbb {H}_{K}^{N}$ $H_{K}^{N}$ of constant curvature K if $K<0$ $K < 0$ ). 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. 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 $\mathbb {H}_{K}^{N},$ $H_{K}^{N},$ in order to construct an isometric immersion of a weighted Fermat-Frechet multitree in the K-space. Finally, we create a new variational method, which differs from Schlafli’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 gives the weighted Fermat-Frechet solution for a sextuple of edge lengths determining (Frechet) tetrahedra.