Geometric features of Vessiot--Guldberg Lie algebras of conformal and Killing vector fields on $\mathbb{R}^2$

TL;DR

This paper classifies finite-dimensional Lie algebras of conformal and Killing vector fields on ^2 relative to arbitrary pseudo-Riemannian metrics, exploring their geometric properties and illustrating with physical examples.

Contribution

It provides a local classification of these Lie algebras and analyzes their geometric features, such as invariant distributions and symplectic structures.

Findings

Classification of Lie algebras of conformal and Killing vector fields on ^2

Analysis of their geometric properties including invariant distributions

Application to physical equations like Milne--Pinney and projective Schrd6dinger

Abstract

This paper locally classifies finite-dimensional Lie algebras of conformal and Killing vector fields on $\mathbb{R}^2$ relative to an arbitrary pseudo-Riemannian metric. Several results about their geometric properties are detailed, e.g. their invariant distributions and induced symplectic structures. Findings are illustrated with two examples of physical nature: the Milne--Pinney equation and the projective Schr\"odinger equation on the Riemann sphere.