Metric‐preserving mappings

Abstract

It is shown that the set F of bijective mappings on a metric space which preserve the metric, forms a group under composition. A metric don Rⁿ is developed using continuous bijective functions g ₁ ... ,g_n on R, it is shown that dinduces the usual topology on Rⁿ. In R¹ with the metric d,the group F_g of metric preserving bijective mappings is shown to be isomorphic to the group F₍ of bijective mappings in R which preserve the usual metric or a subgroup of F ₁ if the function g ₁ is not surjective. A method of generating metrics from an existing metric is introduced. It is shown that these metrics induce the same topology on Rⁿ and the group of metric preserving bijective mappings is isomorphic to the group F_n of metric preserving mappings in Rⁿ with the same topology. If the metric on Rⁿ is induced by some norm, then it is shown that the group of bijective mappings which preserve the above metric is isomorphic to a subgroup of F_n. It is seen that Ulam's conjecture ‘if dis a metric on Rⁿ which induces the usual topology on Rⁿ, then the set of bijective mappings in Rⁿ which preserve dis a group under composition and is isomorphic to a subgroup of the group of mappings which preserve the usual metric’, holds in the cases studied above.