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Contact Lie systems on Riemannian and Lorentzian spaces: from scaling symmetries to curvature-dependent reductions

Rutwig Campoamor-Stursberg, Oscar Carballal, Francisco J. Herranz

TL;DR

The paper develops a curvature-aware framework that generalizes scaling reductions of LH systems to contact Lie systems on 3D Cayley–Klein spaces. It shows how LH systems on symplectic manifolds, when endowed with scaling symmetries, reduce to contact LH systems on CK spaces such as $bS^3$ and $ ext{AdS}^{2+1}$, with projections to lower-dimensional LH systems on spaces like $bS^2$ and $bR^2$. It provides explicit constructions in the time-dependent harmonic oscillator and thermodynamics, and then embeds these into the CK taxonomy, yielding a broad family of contact Lie systems on $3$D CK spaces, including Liouville-type subsystems and curvature-dependent principal bundles. The results connect contact geometry, Sasakian structures, and CK geometry, offering a geometric pathway to new higher-dimensional Lie systems and potential applications in 3D gravity and thermodynamics.

Abstract

We propose an adaptation of the notion of scaling symmetries for the case of Lie-Hamilton systems, allowing their subsequent reduction to contact Lie systems. As an illustration of the procedure, time-dependent frequency oscillators and time-dependent thermodynamic systems are analyzed from this point of view. The formalism provides a novel method for constructing contact Lie systems on the three-dimensional sphere, derived from recently established Lie-Hamilton systems arising from the fundamental four-dimensional representation of the symplectic Lie algebra $\mathfrak{sp}(4,\mathbb{R})$. It is shown that these systems are a particular case of a larger hierarchy of contact Lie systems on a special class of three-dimensional homogeneous spaces, namely the Cayley-Klein spaces. These include Riemannian spaces (sphere, hyperbolic and Euclidean spaces), pseudo-Riemannian spaces (anti-de Sitter, de Sitter and Minkowski spacetimes), as well as Newtonian or non-relativistic spacetimes. Under certain topological conditions, some of these systems retrieve well-known two-dimensional Lie-Hamilton systems through a curvature-dependent reduction.

Contact Lie systems on Riemannian and Lorentzian spaces: from scaling symmetries to curvature-dependent reductions

TL;DR

The paper develops a curvature-aware framework that generalizes scaling reductions of LH systems to contact Lie systems on 3D Cayley–Klein spaces. It shows how LH systems on symplectic manifolds, when endowed with scaling symmetries, reduce to contact LH systems on CK spaces such as and , with projections to lower-dimensional LH systems on spaces like and . It provides explicit constructions in the time-dependent harmonic oscillator and thermodynamics, and then embeds these into the CK taxonomy, yielding a broad family of contact Lie systems on D CK spaces, including Liouville-type subsystems and curvature-dependent principal bundles. The results connect contact geometry, Sasakian structures, and CK geometry, offering a geometric pathway to new higher-dimensional Lie systems and potential applications in 3D gravity and thermodynamics.

Abstract

We propose an adaptation of the notion of scaling symmetries for the case of Lie-Hamilton systems, allowing their subsequent reduction to contact Lie systems. As an illustration of the procedure, time-dependent frequency oscillators and time-dependent thermodynamic systems are analyzed from this point of view. The formalism provides a novel method for constructing contact Lie systems on the three-dimensional sphere, derived from recently established Lie-Hamilton systems arising from the fundamental four-dimensional representation of the symplectic Lie algebra . It is shown that these systems are a particular case of a larger hierarchy of contact Lie systems on a special class of three-dimensional homogeneous spaces, namely the Cayley-Klein spaces. These include Riemannian spaces (sphere, hyperbolic and Euclidean spaces), pseudo-Riemannian spaces (anti-de Sitter, de Sitter and Minkowski spacetimes), as well as Newtonian or non-relativistic spacetimes. Under certain topological conditions, some of these systems retrieve well-known two-dimensional Lie-Hamilton systems through a curvature-dependent reduction.

Paper Structure

This paper contains 26 sections, 14 theorems, 178 equations, 2 figures, 6 tables.

Table of Contents

  1. Introduction
  2. Reduction of Lie--Hamilton systems by scaling symmetries
  3. Application to the harmonic oscillator with time-dependent frequency
  4. Application to a time-dependent thermodynamic system
  5. Contact Lie systems on the sphere $\mathbf{S}^{3}$ from scaling symmetries
  6. Three-dimensional Cayley--Klein spaces
  7. Cayley--Klein Lie algebras and groups
  8. Cayley--Klein spaces and coordinate systems
  9. Contact structure on Cayley--Klein spaces and principal bundles
  10. Positive curvature: $\mathbf{S}^{1}$-principal bundles
  11. Principal bundle $\mathbf{S}^{1} \hookrightarrow \mathbf{A}\mathbf{d}\mathbf{S}^{2+1} \to \mathbb{C} \mathbf{H}^{1}$
  12. Principal bundle $\mathbf{S}^{1} \hookrightarrow \mathbf{N}\mathbf{H}_{+}^{2+1} \to \mathbb{R}^{2}$
  13. Nonpositive curvature: $\mathbb{R}$-principal bundles
  14. Principal bundle $\mathbb{R} \hookrightarrow \{ \mathbf{E}^{3}, \mathbf{G}^{2+1}, \mathbf{M}^{2+1} \} \to \mathbb{R}^{2}$
  15. Principal bundle $\mathbb{R} \hookrightarrow \mathbf{H}^{3} \to \mathbb{R}^{2}$
  16. ...and 11 more sections

Key Result

Lemma 2.3

Let $\Phi: \mathbb{R}^{\times} \times M \to M$ be a principal $\mathbb{R}^{\times}$-action on a symplectic manifold $(M, \omega)$ such that $\omega$ is $1$-homogeneous. Then,

Figures (2)

  • Figure 1: The CK Lie algebras $\mathfrak{s}\mathfrak{o}_{\bm{\kappa}}(4)$ with commutation relations \ref{['eq:3DCK:commrel_Cartan']}. Each point corresponds to a set of values of the graded contraction parameters $\bm{\kappa}=(\kappa_{1}, \kappa_{2}, \kappa_{3})$ with $\kappa_{m} \in \left\{ +, 0, - \right\}$ and the semisimple algebras placed at the vertices. An Inönü--Wigner contraction of the parameter $\kappa_{m} \to 0$ moves from the sides to the centre slice of the 'cube' following the direction $\kappa_{m}$.
  • Figure 2: The three 1D CK spaces $\mathbf{S}^{1}_{[\kappa_1]}$ according to the normalized values of $\kappa_1 \in \left\{ +1,0,-1 \right\}$ and their identifications as real 2D Lie groups. $\mathrm{SO}^{+}(1,1)$ denotes the identity component of $\mathrm{SO}(1,1)$.

Theorems & Definitions (23)

  • Definition 2.1
  • Definition 2.2
  • Remark
  • Lemma 2.3: Libermann1987
  • Proposition 2.4
  • Corollary 2.5
  • Proposition 2.6
  • proof
  • Proposition 2.7: Bravetti2024
  • Theorem 2.8
  • ...and 13 more