Special Conformal Groups of a Riemannian Manifold and Lie Point Symmetries of the Nonlinear Poisson Equation

TL;DR

This paper classifies Lie point symmetries of nonlinear Poisson equations on (pseudo) Riemannian manifolds, linking them to conformal groups and deriving conservation laws, with applications to Thurston geometries.

Contribution

It provides a complete classification of symmetries for nonlinear Poisson equations on generic manifolds and explores their geometric and physical implications.

Findings

Lie symmetries project onto conformal subgroups of the manifold

Critical nonlinearity symmetries correspond to the conformal group when scalar curvature vanishes

Results are applied to Thurston geometries

Abstract

We obtain a complete group classification of the Lie point symmetries of nonlinear Poisson equations on generic (pseudo) Riemannian manifolds M. Using this result we study their Noether symmetries and establish the respective conservation laws. It is shown that the projection of the Lie point symmetries on $M$ are special subgroups of the conformal group of M. In particular, if the scalar curvature of M vanishes, the projection on M of the Lie point symmetry group of the Poisson equation with critical nonlinearity is the conformal group of the manifold. We illustrate our results by applying them to the Thurston geometries.