Constant Negative Curvature Research Articles

Asymptotic relative equilibrium in the n-body problem: Relativistic application in the Poincaré upper half-plane

In this paper, we study the [Formula: see text]-body problem in the Poincaré upper half-plane [Formula: see text], where the radius [Formula: see text] of the Poincaré disk is fixed. We introduce a new potential to derive the condition for hyperbolic relative equilibria on [Formula: see text]. We analyze the relative equilibrium of positive masses moving along geodesics under the [Formula: see text] group. This result is utilized to establish the existence of relative equilibria for the [Formula: see text]-body problem on [Formula: see text] for [Formula: see text] and [Formula: see text]. We revisit previously known results and uncover new qualitative findings on relative equilibria that are not evident in an extrinsic context. Additionally, we provide a simple expression for the center of mass of a system of point particles on a two-dimensional surface with negative constant Gaussian curvature.

Superintegrability, symmetry and point particle T-duality

We show that the ideas related to integrability and symmetry play an important role not only in the string T-duality story but also in its point particle counterpart. Applying those ideas, we find that the T-duality seems to be a more widespread phenomenon in the context of the point particle dynamics than it is in the string one; moreover, it concerns physically very relevant point particle dynamical systems and not just somewhat exotic ones fabricated for the purpose. As a source of T-duality examples, we consider maximally superintegrable spherically symmetric electro-gravitational backgrounds in [Formula: see text] dimensions. We then describe in detail four such spherically symmetric dynamical systems which are all mutually interconnected by a web of point particle T-dualities. In particular, the dynamics of a charged particle scattered by a repulsive Coulomb potential in a flat space is T-dual to the dynamics of the Coulomb scattering in the space of constant negative curvature, but it is also T-dual to the (conformal) Calogero–Moser inverse square dynamics both in flat and hyperbolic spaces. Thus knowing just the Hamiltonian dynamics of the scattered particle cannot give us information about the curvature of the space in which the particle moves.

Generalized Willmore energies, Q-curvatures, extrinsic Paneitz operators, and extrinsic Laplacian powers

Over forty years ago, Paneitz, and independently Fradkin and Tseytlin, discovered a fourth-order conformally invariant differential operator, intrinsically defined on a conformal manifold, mapping scalars to scalars. This operator is a special case of the so-termed extrinsic Paneitz operator defined in the case when the conformal manifold is itself a conformally embedded hypersurface. In particular, this encodes the obstruction to smoothly solving the five-dimensional scalar Laplace equation, and suitable higher dimensional analogs, on conformally compact structures with constant scalar curvature. Moreover, the extrinsic Paneitz operator can act on tensors of general type by dint of being defined on tractor bundles. Motivated by a host of applications, we explicitly compute the extrinsic Paneitz operator. We apply this formula to obtain: an extrinsically-coupled [Formula: see text]-curvature for embedded four-manifolds, the anomaly in renormalized volumes for conformally compact five-manifolds with negative constant scalar curvature, Willmore energies for embedded four-manifolds, the local obstruction to smoothly solving the five-dimensional singular Yamabe problem, and new extrinsically-coupled fourth- and sixth-order operators for embedded surfaces and four-manifolds, respectively.

Fermion-antifermion pair in magnetized optical wormhole background

We are interested in a fermion-antifermion pair in magnetized 2+1 dimensional optical background with constant negative curvature. One implements this background for obtaining the optical metric of the BTZ black hole similar to the hyperbolic wormhole. To analyze the evolution of the considered system, we start with constructing the corresponding form of a one-time fully-covariant many-body Dirac equation derived from quantum electrodynamics. The associated equation leads 4×4 dimensional matrix equation resulting in a set of coupled equations consisting of four first-order equations since the two-body equation possesses spin algebra constructed through the Kronecker product of the generalized Dirac matrices. Also, we show a nonperturbative wave equation for the considered system. Accordingly, we seek analytical solutions to the wave equation. We obtain that relativistic energy of the system depends on the parameters of the background as well as the magnetic field. Also, we adapt the results to a Weyl pair. Our results imply that the lifetime of such a pair can be actively tuned, in principle.

Caustics of pseudo-spherical surfaces in the Euclidean 3-space

Abstract We study geometric properties of caustics of pseudo-spherical surfaces, that is, surfaces with constant negative Gaussian curvature -1 in the Euclidean 3-space ℝ 3 {\mathbb{R}^{3}} . We investigate the Gaussian and the mean curvature of caustics of pseudo-spherical surfaces. Moreover, a certain condition required for the caustics to be minimal surfaces is derived.

Ginzburg-Landau equations on non-compact Riemann surfaces

We study the Ginzburg-Landau equations on line bundles over non-compact Riemann surfaces with constant negative curvature. We prove existence of solutions with energy strictly less than that of the constant curvature (magnetic field) one. These solutions are the non-commutative generalizations of the Abrikosov vortex lattice of superconductivity. Conjecturally, they are (local) minimizers of the Ginzburg-Landau energy. We obtain precise asymptotic expansions of these solutions and their energies in terms of the curvature of the underlying Riemann surface. Among other things, our result shows the spontaneous breaking of the gauge-translational symmetry of the Ginzburg-Landau equations.

On the Hyperbolic Bloch Transform

Motivated by recent theoretical and experimental developments in the physics of hyperbolic crystals, we study the noncommutative Bloch transform of Fuchsian groups that we call the hyperbolic Bloch transform. First, we prove that the hyperbolic Bloch transform is injective and “asymptotically unitary” already in the simplest case—that is, when the Hilbert space is the regular representation of the Fuchsian group, $$\Gamma $$ . Second, when $$\Gamma \subset \textrm{PSU} (1, 1)$$ acts isometrically on the hyperbolic plane $$\mathbb {H}$$ and the Hilbert space is $$L^2 \left( \mathbb {H} \right) $$ , we define a modified, geometric Bloch transform that sends wave functions to sections of irreducible, flat, Hermitian vector bundles over $$\Sigma = \mathbb {H} / \Gamma $$ and transforms the hyperbolic Laplacian into the covariant one.

Reversible Finsler metrics of constant flag curvature

Without the restriction of quadratic form as a Riemannian metric, a Finsler metric F=F(x,y) on a smooth manifold M can be reversible (symmetric in y) or not. Reversible Finsler metrics have different properties from Riemannian metrics though it seems they are very close to Riemannian metrics. Hilbert metric is the famous reversible Finsler metric of negative constant flag curvature in the history, and it is projectively flat. Then it is natural to ask the question how to classify reversible projectively flat Finsler metrics of constant flag curvature and give more new examples? In this paper, we answer the above question by giving the classification when the flag curvature K=−1,0,1 respectively. Especially, for the case when K=−1, we show that the only reversible projectively flat Finsler metrics are just Hilbert metrics. For the case when K=1, we give an algebraic way to construct explicit metric function by solving algebraic equations, such as by solving a quartic equation. When the flag curvature is zero, it is much easier to construct reversible projectively flat Finsler metrics than before.

Graph Surfaces Invariant by Parabolic screw Motions with Constant Curvature in $ \: \mathbb H^2 \times \mathbb R$

In this work we study vertical graph surfaces invariant by parabolic screw motions with pitch $\ell >0$ and constant Gaussian curvature or constant extrinsic curvature in the product space $\mathbb H^2 \times \mathbb R$. In particular, we determine flat and extrinsically flat graph surfaces in $\mathbb H^2 \times \mathbb R$. We also obtain complete and non-complete vertical graph surfaces in $\mathbb H^2 \times \mathbb R$ with negative constant Gaussian curvature and zero extrinsic curvature.

Surfaces with Constant Negative Curvature

In this paper, we have considered surfaces with constant negative Gaussian curvature in the simply isotropic 3-Space by defined Sauer and Strubeckerr. Firstly, we have studied the isotropic II-flat, isotropic minimal and isotropic II-minimal, the constant second Gaussian curvature, and the constant mean curvature of surfaces with constant negative curvature (SCNC) in the simply isotropic 3-space. Surfaces with symmetry are obtained when the mean curvatures are equal. Further, we have investigated the constant Casorati, the tangential and the amalgamatic curvatures of SCNC.

Theoretical studies on optical properties of Beltrami-shaped curved graphene

We compute the optical conductivity and polarization for an out-of-plane deformation in graphene nanostructure using a theoretical approach based on Dirac equation solutions on curved dimensional space-time, where the space part is considered to correspond to the Beltrami pseudosphere which belongs to the family of surfaces with negative constant Gaussian curvature. We found that different parameters of deformation along one direction translate into an enhancement of the optical conductivity peaks and polarization magnitude in the far-infrared frequencies. This allows for a very high degree of polarization with a single layer graphene and opens up a potential prospect of employing graphene layer as efficient polarizers. Therefore the experimental predictions related to the electronic configuration of the corresponding graphene-like sample may be explicitly worked out.

Engineering boundary-dominated topological states in defective hyperbolic lattices

Topological matters with lattice disclinations have been widely investigated in Euclidean space. In recent years, exotic topological phases in hyperbolic lattices, which are regular tessellations in the curved space with a constant negative curvature, have been theoretically proposed and experimentally observed, while the investigation of topologically defective hyperbolic lattices is still lacking. Here, we study topological states in the hyperbolic lattice with polygonal defects. It is shown that the topologically one-way boundary state can still exist in the defective hyperbolic Haldane model. Interestingly, we find that only a single bulk site can induce the formation of boundary-dominated one-way propagations in defective hyperbolic lattices. In experiments, we fabricate the defective hyperbolic circuit with a single bulk site to detect the boundary-dominated topological state. Frequency-dependent impedance responses clearly illustrate the existence of nontrivial band gaps and midgap topological boundary states. Furthermore, the backscattering-immune propagation protected by a single bulk site is observed by measuring the dynamics of voltage packet. Our work suggests a useful platform to study topological phases in defective hyperbolic lattices, and may have potential applications in designing high-efficient topological devices with an extremely small bulk region.

Dimension matters when modeling network communities in hyperbolic spaces.

Over the last decade, random hyperbolic graphs have proved successful in providing geometric explanations for many key properties of real-world networks, including strong clustering, high navigability, and heterogeneous degree distributions. These properties are ubiquitous in systems as varied as the internet, transportation, brain or epidemic networks, which are thus unified under the hyperbolic network interpretation on a surface of constant negative curvature. Although a few studies have shown that hyperbolic models can generate community structures, another salient feature observed in real networks, we argue that the current models are overlooking the choice of the latent space dimensionality that is required to adequately represent clustered networked data. We show that there is an important qualitative difference between the lowest-dimensional model and its higher-dimensional counterparts with respect to how similarity between nodes restricts connection probabilities. Since more dimensions also increase the number of nearest neighbors for angular clusters representing communities, considering only one more dimension allows us to generate more realistic and diverse community structures.

Yamabe Solitons and τ-Quasi Yamabe Gradient Solitons on Riemannian Manifolds Admitting Concurrent-Recurrent Vector Fields

Abstract We consider a Riemannian manifold (M,g) admitting a concurrent-recurrent vector field for which the metric g is a Yamabe soliton or a τ-quasi Yamabe gradient soliton. We show that if the metric of a Riemannian three-manifold (M, g) admitting a concurrent-recurrent vector field is a Yamabe soliton, then M is of constant negative curvature –α 2. In this case, we see that the potential vector field is Killing. Next, we show that if the metric of a Riemannian manifold M admitting concurrent-recurrent vector field is a non-trivial r-quasi Yamabe gradient soliton with potential function f, then M has constant scalar curvature and is equal to –n(n – 1)α 2. Finally, an illustrative example is presented.

Effective de Sitter space, quantum behaviour and large-scale spectral dimension (3+1)

De Sitter space-time, essentially our own universe, is plagued by problems at the quantum level. Here we propose that Lorentzian de Sitter space-time is not fundamental but constitutes only an effective description of a more fundamental quantum gravity ground state. This cosmological ground state is a graph, appearing on large scales as a Riemannian manifold of constant negative curvature. We model the behaviour of matter near this equilibrium state as Brownian motion in the effective thermal environment of graph fluctuations, driven by a universal time parameter. We show how negative curvature dynamically induces the asymptotic emergence of relativistic coordinate time and of leading ballistic motion governed by the isometry group of an “effective Lorentzian manifold” of opposite, positive curvature, i.e. de Sitter space-time: free fall in positive curvature is asymptotically equivalent to the leading behaviour of Brownian motion in negative curvature. The local limit theorem for negative curvature implies that the large-scale spectral dimension of this “effective de Sitter space-time” is (3+1) independently of its microscopic topological dimension. In the effective description, the sub-leading component of asymptotic Brownian motion becomes Schrödinger quantum behavior on a 3D Euclidean manifold.

Characterizing candidates for Cannon's conjecture from geometric measure theory

AbstractWe show that recent work of Song implies that torsion‐free hyperbolic groups with Gromov boundary arerealized as fundamental groups of closed 3‐manifolds of constant negative curvature if and only if the solution to an associated spherical Plateau problem for group homology is isometric to such a 3‐manifold, and suggest some related questions.

ON SOME MANIFOLDS OF CONSTANT NEGATIVE CURVATURE

Manifold of constant negative curvature a great role in the field of physics, mathematics and engineering because it paves to the knowledge Gaussian curvature , n-sphere(s) is a topological n-manifold and objects of constant negative curvature are less familiar but they do appear in nature in the shape of corals and leave . No surprisingly , its plays an important role in geometric topology. The study aims to explain a generalization manifold of constant negative curvature . WE followed the analytical induction mathematical method . We found the following some results. Manifold of constant negative curvature indicates to know the behavior of some of the functions and also it reveals the Cartan-Hadamard theorem which is considered one of the importance aims of simply-connected manifold of nonpositive sectional curvature.

Преобразование Бианки псевдосферы

The work is devoted to the study of the Bianchi transform for surfa­ces of constant negative Gaussian curvature. The surfaces of rotation of cons­tant negative Gaussian curvature are the Minding top, the Minding coil, the pseudosphere (Beltrami surface). Surfaces of constant negative Gaus­sian curvature also include Kuens surface and the Dinis surface. The study of surfaces of constant negative Gaussian curvature (pseudosphe­ri­cal surfaces) is of great importance for the interpretation of Lobachevsky planimetry. The connection of the geometric characteristics of pseudos­phe­rical surfaces with the theory of networks, with the theory of solitons, with nonlinear differential equations and sin-Gordon equations is estab­li­shed. The sin-Gordon equation plays an important role in modern physics. Bianchi transformations make it possible to obtain new pseudospherical surfaces from a given pseudospherical surface. The Bianchi transform for the pseudosphere is constructed. Using a mathematical package, the pseu­dos­phere and its Bianchi transform are constructed.

К динамике твердого стержня в пространстве Лобачевского

An absolutely solid infinitely thin rectilinear immaterial finite length rod helical motion problem is considered. Motion is heical relative to the inertia center. Located at rod ends point masses are identiall. The rod movement occurs in the constant negative curvature space (Lobachevsky space) under the gyroscopic forces system influence and a constant tracking force. The gyroscopic forces structure is set with using special conditions, which contain six characteristic constant parameters (gyroscopic coefficients). The characteristic motion properties are given. Components analytical dependense on the kinematic rod screw, rod orientation parameters is founded. Expressions of these parameters as time function, are obtained.

Fourier expansions of vector-valued automorphic functions with non-unitary twists

We provide Fourier expansions of vector-valued eigenfunctions of the hyperbolic Laplacian that are twist-periodic in a horocycle direction. The twist may be given by any endomorphism of a finite-dimensional vector space; no assumptions on invertibility or unitarity are made. Examples of such eigenfunctions include vector-valued twisted automorphic forms of Fuchsian groups. We further provide a detailed description of the Fourier coefficients and explicitly identify each of their constituents, which intimately depend on the eigenvalues of the twisting endomorphism and the size of its Jordan blocks. In addition, we determine the growth properties of the Fourier coefficients.