Introducing a new intrinsic metric

TL;DR

This paper introduces the $t$-metric, an intrinsic metric, and explores its properties, inequalities with other hyperbolic metrics, and behavior under conformal and quasiconformal mappings, supported by analytical and graphical comparisons.

Contribution

The paper presents the $t$-metric and establishes sharp inequalities relating it to existing hyperbolic metrics, along with analysis of its behavior under conformal mappings.

Findings

Sharp inequalities between $t$-metric and hyperbolic metrics

Behavior of $t$-metric under conformal and quasiconformal maps

Graphical and analytical comparison of metric balls

Abstract

A new intrinsic metric called $t$-metric is introduced. Several sharp inequalities between this metric and the most common hyperbolic type metrics are proven for various domains $G\subsetneq\mathbb{R}^n$. The behaviour of the new metric is also studied under a few examples of conformal and quasiconformal mappings, and the differences between the balls drawn with all the metrics considered are compared by both graphical and analytical means.