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

TL;DR

This paper explores two different invariant notions of arc length for curves on surfaces in equiaffine space, analyzing conditions under which they coincide, thus extending classical metric concepts to a more general geometric setting.

Contribution

It introduces and compares two invariant arc length notions in equiaffine geometry and characterizes when they are equivalent, a novel extension of metric concepts beyond Euclidean space.

Findings

Derived necessary and sufficient conditions for the two arc length notions to agree.

Provided examples illustrating when the two arc length functions coincide or differ.

Extended classical metric notions to the setting of equiaffine geometry.

Abstract

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.