On certain symmetries of $\mathbb R^3$ with a diagonal metric
Adara M. Blaga
TL;DR
The paper addresses the problem of characterizing Killing vector fields on $(\mathbb{R}^3,g)$ where $g$ is diagonal with Lamé coefficients $f_i$, by deriving the Levi-Civita connection in the orthonormal frame $E_k=f_k\partial_{x^k}$ and translating the Killing condition into a precise system of PDEs for the frame components $V^k$ of a Killing field $V=\sum V^kE_k$. It then analyzes various natural choices for the $f_i$, yielding necessary and sufficient conditions for the basis fields to be Killing and providing an explicit classification of Killing fields in several subcases, including when the $f_i$ depend on single coordinates, are constant, or are separable. The main contributions include explicit forms for the components $V^k$ under these cases—highlighting affine, polynomial, trigonometric, and exponential families—together with concrete examples of Killing vectors in particular diagonal metrics. These results extend known 2D diagonal-metric isometry analyses to 3D and establish a framework for further exploration of more general diagonal metrics and their isometry algebras. The work has potential implications for understanding symmetries in anisotropic spaces and could guide future classifications of Killing fields for broader Lamé coefficient structures.
Abstract
We determine Killing vector fields on the $3$-dimensional space $\mathbb R^3$ endowed with a special diagonal metric.