标题： 两条弧长的叙述：等仿射空间中曲面曲线的度量概念 显示英文标题

标题： A Tale of Two Arc Lengths: Metric notions for curves in surfaces in equiaffine space

摘要： 在欧几里得几何中，所有度量概念（曲线的弧长，曲面的第一基本形式等）都来源于切向量上的欧几里得内积，而该内积被欧几里得空间的完整对称群（平移、旋转和反射）所保持。 在仿射等距几何中，不存在一个由仿射等距对称群所保持的切向量上的不变内积概念。 然而，仍然可以为非退化曲线定义一个不变的弧长概念，并为仿射空间中的非退化曲面定义一个不变的第一基本形式。 这导致了曲线上包含在曲面中的两种可能的弧长概念，这两种弧长函数不一定一致。 在本文中，我们将推导出这两种弧长函数一致的充要条件，并通过例子进行说明。 显示英文摘要

摘要： In Euclidean geometry, all metric notions (arc length for curves, the first fundamental form for surfaces, etc.) are derived from the Euclidean inner product on tangent vectors, and this inner product is preserved by the full symmetry group of Euclidean space (translations, rotations, and reflections). In equiaffine geometry, there is no invariant notion of inner product on tangent vectors that is preserved by the full equiaffine symmetry group. Nevertheless, it is possible to define an invariant notion of arc length for nondegenerate curves, and an invariant first fundamental form for nondegenerate surfaces in equiaffine space. This leads to two possible notions of arc length for a curve contained in a surface, and these two arc length functions do not necessarily agree. In this paper we will derive necessary and sufficient conditions under which the two arc length functions do agree, and illustrate with examples.