Two-dimensional Spaces Of Constant Curvature Research Articles

Three-generation solutions of equations of motion in heterotic supergravity

We study the generation number of massless fermions in compactifications recently found in heterotic supergravity. The internal spaces are products of two-dimensional spaces of constant curvature, and the standard embedding is not assumed. The generation number is constrained by the equations of motion and the Bianchi identity. In the case that the Euler characteristics of the three internal submanifolds are $(2,-2,-2)$, the generation number is three or less. We also show that three-generation solutions exist for arbitrary Euler characteristics of the negatively curved 2-manifolds, and we present some explicit solutions.

Classical defects in higher-dimensional Einstein gravity coupled to nonlinear $$\sigma $$ σ -models

We construct solutions of higher-dimensional Einstein gravity coupled to nonlinear $\sigma$-model with cosmological constant. The $\sigma$-model can be perceived as exterior configuration of a spontaneously-broken $SO(D-1)$ global higher-codimensional "monopole". Here we allow the kinetic term of the $\sigma$-model to be noncanonical; in particular we specifically study a quadratic-power-law type. This is some possible higher-dimensional generalization of the Bariola-Vilenkin (BV) solutions with $k$-global monopole studied recently. The solutions can be perceived as the exterior solution of a black hole swallowing up noncanonical global defects. Even in the absence of comological constant its surrounding spacetime is asymptotically non-flat; it suffers from deficit solid angle. We discuss the corresponding horizons. For $\Lambda>0$ in $4d$ there can exist three extremal conditions (the cold, ultracold, and Nariai black holes), while in higher-than-four dimensions the extremal black hole is only Nariai. For $\Lambda<0$ we only have black hole solutions with one horizon, save for the $4d$ case where there can exist two horizons. We give constraints on the mass and the symmetry-breaking scale for the existence of all the extremal cases. In addition, we also obtain factorized solutions, whose topology is the direct product of two-dimensional spaces of constant curvature ($M_2$, $dS_2$, or $AdS_2$) with (D-2)-sphere. We study all possible factorized channels.

On square-integrability of solutions of the stationary Schrödinger equation for the quantum harmonic oscillator in two dimensional constant curvature spaces

In this work, one provides a justification of the condition that is usually imposed on the parameters of the hypergeometric equation, related to the solutions of the stationary Schrödinger equation for the harmonic oscillator in two-dimensional constant curvature spaces, in order to determine the solutions which are square-integrable. One proves that in case of negative curvature, it is a necessary condition of square integrability and in case of positive curvature, a necessary condition of regularity. The proof is based on the analytic continuation formulas for the hypergeometric function. It is observed also that the same is true in case of a slightly more general potential than the one for harmonic oscillator.

Evolutes and isoperimetric deficit in two-dimensional spaces of constant curvature

We relate the total curvature and the isoperimetric deficit of a curve $\gamma $ in a two-dimensional space of constant curvature with the area enclosed by the evolute of $\gamma $. We provide also a Gauss-Bonnet theorem for a special class of evolutes.

A study of the orthogonal polynomials associated with the quantum harmonic oscillator on constant curvature spaces

Recently, Cariñena, et al. [Ann. Phys. 322, 434 (2007)] introduced a new family of orthogonal polynomials that appear in the wave functions of the quantum harmonic oscillator in two-dimensional constant curvature spaces. They are a generalization of the Hermite polynomials and will be called curved Hermite polynomials in the following. We show that these polynomials are naturally related to the relativistic Hermite polynomials introduced by Aldaya et al. [Phys. Lett. A 156, 381 (1991)], and thus are Jacobi polynomials. Moreover, we exhibit a natural bijection between the solutions of the quantum harmonic oscillator on negative curvature spaces and on positive curvature spaces. At last, we show a maximum entropy property for the ground states of these oscillators.

Quantization of a nonlinear oscillator as a model of the harmonic oscillator on spaces of constant curvature: One- and two-dimensional systems

The quantum version of a nonlinear oscillator, previously analyzed at the classical level, is studied first in one dimension and then in two dimensions. This is a problem of quantization of a system with position-dependent mass of the form m = (1 + λx2)−1 and with a λ-dependent nonpolynomial rational potential. The quantization procedure analyzes the existence of Killing vectors and makes use of an invariant measure. It is proved that this system can be considered as a model of the quantum harmonic oscillator on two-dimensional spaces of constant curvature.

Separable Potentials and a Triality in Two-Dimensional Spaces of Constant Curvature

We characterize and completely describe some types of separable potentials in two-dimensional spaces, , of any (positive, zero or negative) constant curvature and either definite or indefinite signature type. The results are formulated in a way which applies at once for the two-dimensional sphere S2, hyperbolic plane H2, AntiDeSitter / DeSitter two-dimensional spaces AdS1+1 / dS1+1 as well as for their flat analogues E2 and M1+1. This is achieved through an approach of Cayley-Klein type with two parameters, κ1 and κ2, to encompass all curvatures and signature types. We discuss six coordinate systems allowing separation of the Hamilton-Jacobi equation for natural Hamiltonians in and relate them by a formal triality transformation, which seems to be a clue to introduce general “elliptic coordinates” for any CK space concisely. As an application we give, in any , the explicit expressions for the Fradkin tensor and for the Runge-Lenz vector, i.e., the constants of motion for the harmonic oscillator and Kepler potential on any .

Action-Angle Analysis of Some Geometric Models of Internal Degrees of Freedom

We derive and discuss equations of motion of infinitesimal affinely-rigid body moving in Riemannian spaces. There is no concept of extended rigid and affinely rigid body in a general Riemannian space. Therefore the gyroscopes with affine degrees of freedom are described as moving bases attached to the material point. This base is a remnant of extended rigid and affinely rigid body in a flat space. The special stress is laid on affinely rigid bodies in two-dimensional constant curvature spaces (sphere and pseudosphere). In particular, we consider incompressible affinely rigid bodies, like, e.g. fat spots on a water surface (e.g. petrol pollution). This is a two-dimensional analogue of three-dimensional incompressible objects like fluid droplets.

Completeness of superintegrability in two-dimensional constant-curvature spaces

We classify the Hamiltonians H = px2 + py2 + V(x,y) of all classical superintegrable systems in two-dimensional complex Euclidean space with two additional second-order constants of the motion. We similarly classify the superintegrable Hamiltonians H = J12 + J22 + J32 + V(x, y, z) on the complex two-sphere where x2 + y2 + z2 = 1. This is achieved in all generality using properties of the complex Euclidean group and the complex orthogonal group.

Trigonometry of spacetimes: a new self-dual approach to a curvature/signature (in)dependent trigonometry

A new method to obtain trigonometry for the real spaces of constant curvature and metric of any (even degenerate) signature is presented. The method encapsulates trigonometry for all these spaces into a single basic trigonometric group equation. This brings to its logical end the idea of an absolute trigonometry, and provides equations which hold true for the nine two-dimensional spaces of constant curvature and any signature. This family of spaces includes both relativistic and non-relativistic homogeneous spacetimes; therefore a complete discussion of trigonometry in the six de Sitter, minkowskian, Newton--Hooke and galilean spacetimes follow as particular instances of the general approach. Any equation previously known for the three classical riemannian spaces also has a version for the remaining six spacetimes; in most cases these equations are new. Distinctive traits of the method are universality and self-duality: every equation is meaningful for the nine spaces at once, and displays explicitly invariance under a duality transformation relating the nine spaces. The derivation of the single basic trigonometric equation at group level, its translation to a set of equations (cosine, sine and dual cosine laws) and the natural apparition of angular and lateral excesses, area and coarea are explicitly discussed in detail. The exposition also aims to introduce the main ideas of this direct group theoretical way to trigonometry, and may well provide a path to systematically study trigonometry for any homogeneous symmetric space.

Supersymmetric black holes

The effective action of d=4, N=2 supergravity is shown to acquire no quantum corrections in background metrics admitting supercovariantly constant spinors. In particular, these metrics include the Robinson-Bertotti metric (product of two two-dimensional spaces of constant curvature) with all eight supersymmetries unbroken. Another example is a set of arbitrary number of extreme Reissner-Nodström black holes break four of eight supersymmetries, leaving the other four unbroken. We have found manifestly supersymmetric black holes, which are non-trivial solutions of the flatness condition D 2=0 of the corresponding (shortened) superspace. Their bosonic part describes a set of extreme Reissner-Nordström black hole solutions are exact even when all quantum supergravity corrections are taken into account.

Einstein-Maxwell instantons and charge quantization

We describe a class of solutions of the source-free Einstein-Maxwell equations, satisfying two given ansätze; these solutions represent products of two two-dimensional spaces of constant curvature containing a uniform electromagnetic field. Some of these solutions are compact and can be considered as gravitational-electromagnetic instantons. Due to the nontrivial topology of these four-dimensional «space-times», the fluxes of the electric and magnetic field through suitable surfaces (which we call «charges», in the sense of Wheeler) turn out to be independently quantized.