Distortion coefficients of the $\alpha$-Grushin plane

TL;DR

This paper calculates the distortion coefficients of the $lpha$-Grushin plane using generalized trigonometric functions, provides estimates, and proposes a conjecture related to synthetic curvature conditions.

Contribution

It introduces explicit formulas for distortion coefficients of the $lpha$-Grushin plane and suggests a new conjecture on synthetic curvature conditions for these geometries.

Findings

Explicit formulas for distortion coefficients in terms of generalized trigonometric functions.

Derived estimates for the distortion coefficients.

Proposed a conjecture on synthetic curvature conditions.

Abstract

We compute the distortion coefficients of the $\alpha$-Grushin plane. They are expressed in terms of generalised trigonometric functions. Estimates for the distortion coefficients are then obtained and a conjecture of a synthetic curvature condition for the generalised Grushin planes is suggested.