N-dimensional Spaces Of Constant Curvature Research Articles

Integrable generalizations of oscillator and Coulomb systems via action–angle variables

Oscillator and Coulomb systems on N-dimensional spaces of constant curvature can be generalized by replacing their angular degrees of freedom with a compact integrable (N−1)-dimensional system. We present the action–angle formulation of such models in terms of the radial degree of freedom and the action–angle variables of the angular subsystem. As an example, we construct the spherical and pseudospherical generalization of the two-dimensional superintegrable models introduced by Tremblay, Turbiner and Winternitz and by Post and Winternitz. We demonstrate the superintegrability of these systems and give their hidden constant of motion.

Superintegrability on N-dimensional spaces of constant curvature from so(N + 1) and its contractions

The Lie—Poisson algebra so(N + 1) and some of its contractions are used to construct a family of superintegrable Hamiltonians on the N-dimensional spherical, Euclidean, hyperbolic, Minkowskian, and (anti-)de Sitter spaces. We firstly present a Hamiltonian which is a superposition of an arbitrary central potential with N arbitrary centrifugal terms. Such a system is quasi-maximally superintegrable since this is endowed with 2N — 3 functionally independent constants of motion (plus the Hamiltonian). Secondly, we identify two maximally superintegrable Hamiltonians by choosing a specific central potential and finding at the same time the remaining integral. The former is the generalization of the Smorodinsky—Winternitz system to the above six spaces, while the latter is a generalization of the Kepler—Coulomb potential, for which the Laplace—Runge—Lenz N vector is also given. All the systems and constants of motion are explicitly expressed in a unified form in terms of ambient and polar coordinates as they are parametrized by two contraction parameters (curvature and signature of the metric).

A multidimensional radiation-filled universe

We consider a radiation-filled universe which possesses the product symmetry: (N-dimensional space of constant curvature) × (n sphere). The solutions of all the types, within this class, to the classical field equations are given. In the case of theN-dimensional space of zero or negative curvature constant, the solutions exhibit a tendency to approach asymptotically the Kasner-like state in which theN-dimensional subspace expands while then sphere shrinks to the final singularity. Our conclusions based on the phase-diagram method are in agreement with the results concerning the ℝN × Sn universe calculated by Sahdev with the help of numerical methods.

Casimir operators of groups of motions of spaces of constant curvature

Limit transitions are constructed between the generators (Casimir operators) of the center of the universal covering algebra for the Lie algebras of the groups of motions of n-dimensional spaces of constant curvature. A method is proposed for obtaining the Casimir operators of a group of motions of an arbitrary n-dimensional space of constant curvature from the known Casimir operators of the group SO(n + 1). The method is illustrated for the example of the groups of motions of four-dimensional spaces of constant curvature, namely, the Galileo, Poincare, Lobachevskii, de Sitter, Carroll, and other spaces.

Local imbedding of Einstein spaces

A special class of hypersurfaces of a Riemannian space is examined, this class being defined by the stipulation that the coefficients of the third fundamental form be expressible as linear combinations of the coefficients of the first and second fundamental forms. It is jound that these so-called C-hypersurfaces are umbilical provided that certain conditions (which may depend on dimension) are satisfied. An (n-1)-dimensional Einstein space imbedded in an n-dimensional space of constant curvature is such a C-hypersurface; accordingly the theory may be applied to the problem of the local imbedding of such spaces.