曲线上穹顶的共线性

arXiv - MATH - Metric Geometry Pub Date : 2024-08-05 DOI:arxiv-2408.02517

Robert Miranda

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引用次数: 0

摘要

积分曲线是由单位区间组成的封闭的片断线性曲线。穹顶是面为等边三角形、边界为积分曲线的多面体。Glazyrin 和 Pak 通过分析单位菱形的情况，证明并非每条积分曲线都能形成穹顶，并推测每条积分曲线都与单位菱形共线。我们证明了这一推测对于有向圆顶来说是错误的，但每条积分曲线都与有限多个单位菱形的联合体共线。

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Cobordism of domes over curves

An integral curve is a closed piecewise linear curve comprised of unit intervals. A dome is a polyhedral surface whose faces are equilateral triangles and whose boundary is an integral curve. Glazyrin and Pak showed that not every integral curve can be domed by analyzing the case of unit rhombi, and conjectured that every integral curve is cobordant to a unit rhombus. We show that this is false for oriented domes, but that every integral curve is cobordant to the union of finitely many unit rhombi.

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arXiv - MATH - Metric Geometry

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