  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.
