The harmonic oscillator on Riemannian and Lorentzian configuration spaces of constant curvature (original) (raw)

2008, Journal of Mathematical Physics

The harmonic oscillator as a distinguished dynamical system can be defined not only on the Euclidean plane but also on the sphere and on the hyperbolic plane, and more generally on any configuration space with constant curvature and with a metric of any signature, either Riemannian (definite positive) or Lorentzian (indefinite). In this paper we study the main properties of these 'curved' harmonic oscillators simultaneously on any such configuration space, using a Cayley-Klein (CK) type approach, with two free parameters κ1, κ2 which altogether correspond to the possible values for curvature and signature type: the generic Riemannian and Lorentzian spaces of constant curvature (sphere S 2 , hyperbolic plane H 2 , AntiDeSitter sphere AdS 1+1 and DeSitter sphere dS 1+1 ) appear in this family, with the Euclidean and Minkowski spaces as flat limits.