R-separation of variables for the conformally invariant Laplace equation

TL;DR

This paper investigates the conditions under which the conformally invariant Laplace equation can be separated using R-separation on Riemannian manifolds, revealing equivalences with additive separation in certain cases.

Contribution

It determines the conditions for R-separation of the conformally invariant Laplace equation and compares them with additive separation conditions for the Hamilton-Jacobi equation, especially in three dimensions.

Findings

On conformally flat manifolds, both equations separate in the same coordinates.

The conditions for R-separation are explicitly characterized.

The analysis is detailed for three-dimensional manifolds.

Abstract

The conditions for R-separation of variables for the conformally invariant Laplace equation on an n-dimensional Riemannian manifold are determined and compared with the conditions for the additive separation of the null geodesic Hamilton-Jacobi equation. The case of 3-dimensions is examined in detail and it is proven that on any conformally flat manifold the two equations separate in the same coordinates.