Certain symmetries of $\mathbb R^2$ with diagonal metrics

TL;DR

This paper investigates the symmetries of $\,\mathbb{R}^2$ equipped with diagonal Riemannian metrics by identifying Killing vector fields under specific conditions on the metric coefficients.

Contribution

It provides a concrete description of the Killing vector fields for a family of diagonal metrics on $\,\mathbb{R}^2$ under certain restrictions, clarifying their symmetry structures.

Findings

Explicit characterization of Killing vector fields for the given metrics.

Conditions on Lamé coefficients that determine the symmetry types.

Enhanced understanding of geometric symmetries in 2D diagonal metric spaces.

Abstract

We put into light the Killing vector fields on $\mathbb R^2$ endowed with a family of diagonal Riemannian metrics. According to certain restrictions on the Lam\'{e} coefficients, we concretely describe the symmetries of the metric.