Constant Curvature Surfaces Research Articles

Logarithmic spirals on surfaces of constant Gaussian curvature

Logarithmic spirals on surfaces of constant Gaussian curvature

Constant sectional curvature surfaces with a semi-symmetric non-metric connection

Consider the Euclidean space R3 endowed with a canonical semi-symmetric non-metric connection determined by a vector field C∈X(R3). We study surfaces when the sectional curvature with respect to this connection is constant. In case that the surface is cylindrical, we obtain full classification when the rulings are orthogonal or parallel to C. If the surface is rotational, we prove that the rotation axis is parallel to C and we classify all conical rotational surfaces with constant sectional curvature. Finally, for the particular case 12 of the sectional curvature, the existence of rotational surfaces orthogonally intersecting the rotation axis is also obtained.

Random circular billiards on surfaces of constant curvature: Pseudo integrability and mixing

Given a random map [Formula: see text], we define a random billiard map on a surface of constant curvature (Euclidean plane, hyperbolic plane, or the sphere). The Liouville measure is invariant for this billiard map. Finally, we show some dynamical properties such as ergodicity in the case of random circular billiards.

Hofer–Zehnder capacity of magnetic disc tangent bundles over constant curvature surfaces

We compute the Hofer–Zehnder capacity of magnetic disc tangent bundles over constant curvature surfaces. We use the fact that the magnetic geodesic flow is totally periodic and can be reparametrized to obtain a Hamiltonian circle action. The oscillation of the Hamiltonian generating the circle action immediately yields a lower bound of the Hofer–Zehnder capacity. The upper bound is obtained from Lu’s bounds of the Hofer–Zehnder capacity using the theory of pseudo-holomorphic curves. In our case, the gradient spheres of the Hamiltonian H will give rise to the non-vanishing Gromov–Witten invariant.

Conformally flat almost cosymplectic 3-manifolds

In this paper, we consider an almost cosymplectic 3-manifold M such that its scalar curvature is invariant along the Reeb vector field. We prove that if the Reeb vector field of M is harmonic, then M is conformally flat if and only if it is locally isometric to the product ℝ × N 2(c), where N 2(c) is a Kähler surface of constant sectional curvature c. Almost cosymplectic 3-manifolds on which the Reeb vector field satisfies the h-a condition together with conformal flatness are also classified.

Spatial-temporal optical vortex pendulum on a curved surface.

Spatial-temporal optical vortices (STOVs) have recently become the focus of newly structured optical fields. In this paper, their propagation on a 2D curved surface named the constant Gaussian curvature surface (CGCS) is studied. Using the matrix optics approach, we provide the analytical solution of the STOV propagation under the paraxial approximation on the CGCS with positive curvature. One method of creating timers is made possible by the spatiotemporal distribution direction of STOV light intensity, which swings like a pendulum throughout the evolution, in contrast to propagation on a flat surface. This swing, however, stops when the curved surface's curvature radius matches the light's Rayleigh distance. Besides, the transverse orbital angular momentum of STOV is deduced, and we find that the intrinsic and extrinsic OAM periodically exchange, but the total transverse OAM is always zero during the propagation on CGCS. It aids in controlling the transverse extrinsic orbital angular momentum of STOV in nontrivial space.

On discrete constant principal curvature surfaces

It has been recently discovered that a certain class of nanocarbon materials has geometrical properties related to the geometry of discrete surfaces with a pre-constant discrete curvature, based on a discrete surface theory for trivalent graphs proposed in 2017 by Kotani et al. In this paper, with the aim of an application to the nanocarbon materials, we will study discrete constant principal curvature (CPC) surfaces. Firstly, we develop the discrete surface theory on a full 3-ary oriented tree so that we define a discrete analogue of principal directions on them and investigate it. We also construct some interesting examples of discrete constant principal curvature surfaces, including discrete CPC tori.

On rationally integrable planar dual multibilliards and piecewise smooth projective billiards

The billiard flow in a planar domain Ω acts on the tangent bundle TR2|Ω as geodesic flow with reflections from the boundary. It has the trivial first integral: squared modulus of the velocity. Bolotin’s conjecture, now a joint theorem of Bialy, Mironov and the author, deals with those billiards whose flow admits an additional integral that is polynomial in the velocity and whose restriction to the unit tangent bundle is non-constant. It states that (1) if the boundary of such a billiard is C 2-smooth, nonlinear and connected, then it is a conic; (2) if it is piecewise C 2-smooth and contains a nonlinear arc, then it consists of arcs of conics from a confocal pencil and segments of ‘admissible lines’ for the pencil; (3) the minimal degree of the additional integral is either 2, or 4. In 1997 Sergei Tabachnikov introduced projective billiards: planar curves equipped with a transversal line field, which defines a reflection acting on oriented lines and the projective billiard flow. They are common generalization of billiards on surfaces of constant curvature, but in general may have no canonical conserved quantity. In a previous paper the author classified those C 4-smooth connected nonlinear planar projective billiards whose flow admits a non-constant integral that is a rational 0-homogeneous function of the velocity (with coefficients depending on the position): these billiards are called rationally 0-homogeneously integrable. It was shown that: (1) the underlying curve is a conic; (2) the minimal degree of integral is equal to two, if the billiard is defined by a dual pencil of conics; (3) otherwise it can be arbitrary even number. In the present paper we classify piecewise C4-smooth rationally 0-homogeneously integrable projective billiards. Unexpectedly, we show that such a billiard associated to a dual pencil of conics may have integral of minimal degree 2, 4, or 12. For the proof of main results we prove dual results for the so-called dual multibilliards introduced in the present paper.

On quasicomplete k$k$‐surfaces in 3‐dimensional space‐forms

AbstractIn the study of immersed surfaces of constant positive extrinsic curvature in space‐forms, it is natural to substitute completeness for a weaker property, which we here call quasicompleteness. We determine the global geometry of such surfaces under the hypotheses of quasicompleteness. In particular, we show that, for , the only quasicomplete immersed surfaces of constant extrinsic curvature equal to in the 3‐dimensional space‐form of constant sectional curvature equal to are the geodesic spheres. Together with earlier work of the author, this completes the classification of quasicomplete immersed surfaces of constant positive extrinsic curvature in 3‐dimensional space‐forms.

Преобразование Бианки катушки Миндинга

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 Mining 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 non-linear 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 Minding coil is constructed. Using a mathematical package, the Minding coil and its Bianchi transform are constructed.

P → ∞ limit of tachyon correlators in (2, 2p + 1) minimal Liouville gravity from classical Liouville theory

Previously it was suggested, motivated by correspondence with JT gravity, that tachyon correlators in (2, 2p+1) minimal Liouville gravity (MLG) in the p → ∞ (semiclassical) limit should be interpreted as moduli space volumes for constant curvature surfaces with conical defects. In this work we propose that these volumes are associated with Kähler metrics on moduli spaces introduced by Zograf and Takhtajan, for which the classical Liouville action is a Kähler potential. We check this proposal by numerical calculation of these Kähler metrics and associated volumes for the simplest example of genus 0 surface with 4 conical defects, using conformal field theory. A peculiar property of MLG correlators is proportionality to number of conformal blocks in a certain region of parameter space; in a particular limiting case, we check this property for the volumes following from classical Liouville action and thus provide an analytic confirmation of our proposal.

Refocusing of the optical branched flow on a rough curved surface

The phenomenon of branched flow has attracted researchers since its inception, with recent observations of the light branching on soap bubbles. However, previous studies have primarily focused on the flat spacetime, overlooking the effect of surface curvature on branched flows. In this paper, we explore the branched flow phenomenon of light on a rough curved surface called constant Gaussian curvature surfaces (CGCSs). Compared with flat space, a CGCS demonstrates that the first branching point advances due to the focusing effect of the positive curvature of the surface. Furthermore, unlike on flat space, optical branches on curved surfaces do not consistently become chaotic during its transmission in a random potential field. On the contrary, the “entropy” decreases at specific positions, which reveals a sink flow phenomenon following the generation of branched flows. This result highlights the time inversion characteristics of CGCSs. Lastly, we demonstrated that the anomalous entropy reduction is related to the transverse and longitudinal coherence transformations of light. We suppose these efforts would fuel further investigation of the thermodynamic evolution and spatiotemporal inversion of random caustics, as well as their future application in the information transmission of random potentials in curved spacetime.

Electrostatic system with divergence-free Bach tensor and non-null cosmological constant

Abstract We prove that three-dimensional electrostatic manifolds with divergence-free Bach tensor are locally conformally flat, provided that the electric field and the gradient of the lapse function are linearly dependent. Consequently, a three-dimensional electrostatic manifold admits a local warped product structure with a one-dimensional base and a constant curvature surface fiber.

Shape-driven confluent rigidity transition in curved biological tissues

Collective cell motions underlie structure formation during embryonic development. Tissues exhibit emergent multicellular characteristics such as jamming, rigidity transitions, and glassy dynamics, but there remain questions about how those tissue-scale dynamics derive from local cell-level properties. Specifically, there has been little consideration of the interplay between local tissue geometry and cellular properties influencing larger-scale tissue behaviors. Here, we consider a simple two-dimensional computational vertex model for confluent tissue monolayers, which exhibits a rigidity phase transition controlled by the shape index (ratio of perimeter to square root area) of cells, on surfaces of constant curvature. We show that the critical point for the rigidity transition is a function of curvature such that positively curved systems are likely to be in a less rigid, more fluid, phase. Likewise, negatively curved systems (saddles) are likely to be in a more rigid, less fluid, phase. A phase diagram we generate for the curvature and shape index constitutes a testable prediction from the model. The curvature dependence is interesting because it suggests a natural explanation for more dynamic tissue remodeling and facile growth in regions of higher surface curvature. Conversely, we would predict stability at the base of saddle-shaped budding structures without invoking the need for biochemical or other physical differences. This concept has potential ramifications for our understanding of morphogenesis of budding and branching structures.

Deployable strip structures

We introduce the new concept of C-mesh to capture kinetic structures that can be deployed from a collapsed state. Quadrilateral C-meshes enjoy rich geometry and surprising relations with differential geometry: A structure that collapses onto a flat and straight strip corresponds to a Chebyshev net of curves on a surface of constant Gaussian curvature, while structures collapsing onto a circular strip follow surfaces which enjoy the linear-Weingarten property. Interestingly, allowing more general collapses actually leads to a smaller class of shapes. Hexagonal C-meshes have more degrees of freedom, but a local analysis suggests that there is no such direct relation to smooth surfaces. Besides theory, this paper provides tools for exploring the shape space of C-meshes and for their design. We also present an application for freeform architectural skins, namely paneling with spherical panels of constant radius, which is an important fabrication-related constraint.

Iterated Medial Triangle Subdivision in Surfaces of Constant Curvature

Consider a geodesic triangle on a surface of constant curvature and subdivide it recursively into four triangles by joining the midpoints of its edges. We show the existence of a uniform delta >0 such that, at any step of the subdivision, all the triangle angles lie in the interval (delta ,pi -delta ). Additionally, we exhibit stabilising behaviours for both angles and lengths as this subdivision progresses.

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.

On the restricted (N + 1)-body problem on surfaces of constant curvature

In this paper, we consider a restricted (N+1)-body problem on surfaces Mκ2, where the constant κ≠0 is the Gaussian curvature, which by means of a rescaling can be reduced to κ=±1. This problem consists in the study of the dynamics of an infinitesimal mass particle attracted by N primaries of identical masses describing elliptic relative equilibria of the N-body problem on Mκ2, i.e., a solution where the primaries are rotating uniformly with constant angular velocity ω on a fixed parallel of S2 or H2 and placed at the vertices of a regular polygon. In a rotating frame, this problem yields a two degrees of freedom Hamiltonian system. The goal of this paper is to describe analytically some dynamics features for κ=±1. Precisely, we study the relative location of equilibria, obtaining, in particular, that the poles of S2 and vertex of H2 represent equilibrium points for any value of the parameters. Thus, analysis of the nonlinear stability of these equilibria is carried out, as well as two types of bifurcations are detected: Hamiltonian-Hopf and N-bifurcation. Additionally, we prove the existence of a family of Hill's orbits and a family of periodic orbits when the primaries are located near the poles of S2 or the vertex of H2. Finally we prove the existence of KAM 2-tori related to these periodic orbits.

Channel linear Weingarten surfaces in space forms

Channel linear Weingarten surfaces in space forms are investigated in a Lie sphere geometric setting, which allows for a uniform treatment of different ambient geometries. We show that any channel linear Weingarten surface in a space form is isothermic and, in particular, a surface of revolution in its ambient space form. We obtain explicit parametrisations for channel surfaces of constant Gauss curvature in space forms, and thereby for a large class of linear Weingarten surfaces up to parallel transformation.

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.