Conformal compactification and cycle-preserving symmetries of spacetimes

Abstract

The cycle-preserving symmetries for the nine two-dimensional real spaces of constant curvature are collectively obtained within a Cayley-Klein framework. This approach affords a unified and global study of the conformal structure of the three classical Riemannian spaces as well as of the six relativistic and non-relativistic spacetimes (Minkowskian, de Sitter, anti-de Sitter, both Newton-Hooke and Galilean), and gives rise to general expressions holding simultaneously for all of them. Their metric structure and cycles (lines with constant geodesic curvature that include geodesics and circles) are explicitly characterized. The corresponding cyclic (Mobius-like) Lie groups together with the differential realizations of their algebras are then deduced; this derivation is new and much simpler than the usual ones and applies to any homogeneous space in the Cayley-Klein family, whether flat or curved and with any signature. Laplace and wave-type differential equations with conformal algebra symmetry are constructed. Furthermore, the conformal groups are realized as matrix groups acting as globally defined linear transformations in a four-dimensional "conformal ambient space", which in turn leads to an explicit description of the "conformal completion" or compactification of the nine spaces.

43 pages, LaTeX

Cited by in corpus (42)

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• Maximal superintegrability on N-dimensional curved spaces
• The anisotropic oscillator on curved spaces: A new exactly solvable model
• Superintegrability on Three-Dimensional Riemannian and Relativistic Spaces of Constant Curvature
• Three-dimensional gravity and Drinfel'd doubles: spacetimes and symmetries from quantum deformations
• The anisotropic oscillator on the two-dimensional sphere and the hyperbolic plane
• Integrable potentials on spaces with curvature from quantum groups
• Maximally superintegrable Smorodinsky-Winternitz systems on the N-dimensional sphere and hyperbolic spaces
• Homogeneous Nonrelativistic Geometries as Coset Spaces
• The $\kappa$-Newtonian and $\kappa$-Carrollian algebras and their noncommutative spacetimes
• Non-commutative relativistic spacetimes and worldlines from 2+1 quantum (anti-)de Sitter groups
• Erlangen Program at Large-1: Geometry of Invariants
• A (2+1) non-commutative Drinfel'd double spacetime with cosmological constant
• Curvature from quantum deformations
• Lorentzian Snyder spacetimes and their Galilei and Carroll limits from projective geometry
• New quantum (anti)de Sitter algebras and discrete symmetries
• A new integrable anisotropic oscillator on the two-dimensional sphere and the hyperbolic plane
• Lie-Hamilton systems on curved spaces: A geometrical approach
• Twisted (2+1) $\kappa$-AdS Algebra, Drinfel'd Doubles and Non-Commutative Spacetimes
• The harmonic oscillator on Riemannian and Lorentzian configuration spaces of constant curvature
• Superintegrability on the Dunkl oscillator model in three-Dimensional spaces of constant curvature
• Two-Dimensional Conformal Models of Space-Time and Their Compactification
• Contractions, deformations and curvature
• Spectrum Generating Algebras for the free motion in $S^3$
• Cayley-Klein Lie bialgebras: Noncommutative spaces, Drinfel'd doubles and kinematical applications
• The Perlick system type I: from the algebra of symmetries to the geometry of the trajectories
• Clifford Algebras and Possible Kinematics
• Factorization approach to superintegrable systems: Formalism and applications
• Higher-order superintegrable momentum-dependent Hamiltonians on curved spaces from the classical Zernike system
• Superintegrability on N-dimensional spaces of constant curvature from so(N+1) and its contractions
• Clifford Fibrations and Possible Kinematics
• From Lorentzian to Galilean (2+1) gravity: Drinfel'd doubles, quantisation and noncommutative spacetimes
• Superintegrability on sl(2)-coalgebra spaces
• $\kappa$-Galilean and $\kappa$-Carrollian noncommutative spaces of worldlines
• Quantum Deformations and Superintegrable Motions on Spaces with Variable Curvature
• A perspective on the Magic Square and the 'special unitary' realizations of simple Lie algebras
• Contact Lie systems on Riemannian and Lorentzian spaces: from scaling symmetries to curvature-dependent reductions
• Lie-Hamilton systems on Riemannian and Lorentzian spaces from conformal transformations and some of their applications
• Curvature as an integrable deformation
• Cayley-Klein Poisson homogeneous spaces
• Doubled Conformal Compactification
• Geometry preserving numerical methods for physical systems with finite-dimensional Lie algebras