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

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.

arXiv comment22 pages

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• Lie-Hamilton systems on the plane: properties, classification and applications
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Cited by in corpus (4)

• Contact Lie systems
• Multisymplectic structures and invariant tensors for Lie systems
• Geometric models for Lie--Hamilton systems on ${\mathbb{R}}^2$
• Reduction and reconstruction of multisymplectic Lie systems