A group-theoretic property of the Euclidean metric

Abstract

Let d d denote a metric on R n ( n ∈ N ) {{\mathbf {R}}^n}(n \in {\mathbf {N}}) compatible with its Euclidean toplogy, and let I ( d ) I(d) be the group of isometries on R n {{\mathbf {R}}^n} relative to this metric. We show that whenever I ( d ) I(d) includes the group of motions then I ( d ) I(d) is identical with it.