Geometric features of Vessiot–Guldberg Lie algebras of conformal and Killing vector fields on $\mathbb{R}^2$
Volume 113 / 2017
Banach Center Publications 113 (2017), 243-262
MSC: Primary 17B66; Secondary 34A26, 53B30, 53B50.
DOI: 10.4064/bc113-0-13
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 Schrödinger equation on the Riemann sphere.
