Properties of isometrically homogeneous curves

TL;DR

This paper investigates isometrically homogeneous spaces homeomorphic to lines, focusing on translation-invariant metrics on real lines, and establishes key equivalences between linear connectedness and Nagata dimension.

Contribution

It characterizes the properties of isometrically homogeneous curves, proving the equivalence between linear connectedness and 1-dimensional Nagata dimension, and provides examples of pathological cases.

Findings

Linear connectedness is equivalent to 1-dimensional Nagata dimension.

Characterization of translation-invariant metrics on the real line.

Examples of pathological isometrically homogeneous curves.

Abstract

This paper is devoted to the study of isometrically homogeneous spaces from the view point of metric geometry. Mainly we focus on those spaces that are homeomorphic to lines. One can reduce the study to those distances on $\R$ that are translation invariant. We study possible values of various metric dimensions of such spaces. One of the main results is the equivalence of two properties: the first one is linear connectedness and the second one is 1-dimensionality, with respect to Nagata dimension. Several concrete pathological examples are provided.