Length Preserving Maps

Abstract

A map 𝜙 of a metric space R in which any two points can be connected by a rectifiable curve into a metric space R’ preserves length or is equilong, if the length of any curve x(t), α≦t≦β, in R equals that of its image 𝜙x(t) in R’. For R = R’ we speak of an equilong map of R. Local isometries are special equilong maps. An equilong map is proper if it is not an isometry.