Constant Negative Gaussian Curvature Research Articles

Superposition Formulae for the Geometric Bäcklund Transformations of the Hyperbolic and Elliptic Sine-Gordon and Sinh-Gordon Equations

We provide superposition formulae for the six cases of Bäcklund transformations corresponding to space-like and time-like surfaces in the 3-dimensional pseudo-Euclidean space. In each case, the surfaces have constant negative or positive Gaussian curvature and they correspond to solutions of one of the following equations: the sine-Gordon, the sinh-Gordon, the elliptic sine-Gordon and the elliptic sinh-Gordon equation. The superposition formulae provide infinitely many solutions algebraically after the first integration of the Bäcklund transformation. Such transformations and the corresponding superposition formulae provide solutions of the same hyperbolic equation, while they show an unusual property for the elliptic equations. The Bäcklund transformation alternates solutions of the elliptic sinh-Gordon equation with those of the elliptic sine-Gordon equation and the superposition formulae provide solutions of the same elliptic equation. Explicit examples and illustrations are given.

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.

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.

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.

The intriguing mechanics of a tractrix of cards

Figure 1 shows the photograph of a tractrix made of cards. This is a simple yet captivating way to make a tractrix — an arrangement that has recently appeared in some engaging pedagogical resources and literature [1, 2]. When closely spaced cards lean like this, one after another, over a horizontal plane, the contour created is a tractrix — a curve of significance in mathematics, since its revolution around its asymptote produces a pseudosphere: a curved surface with constant negative Gaussian curvature [3, 4].

CHARACTERIZATIONS OF NORMAL AND BINORMAL SURFACES IN G3

In this paper, our aim is to give surfaces in the Galilean 3-space G3 with the property that there exist four geodesics through each point such that every surface built with the normal lines and the binormal lines along these geodesics is a surface with a minimal surface and a constant negative Gaussian curvature. We show that should be an isoparametric surface in G3: A plane or a circular hyperboloid.

Wave maps and constant curvature surfaces: singularities and bifurcations

Wave maps (or Lorentzian-harmonic maps) from a $1+1$-dimensional Lorentz space into the $2$-sphere are associated to constant negative Gaussian curvature surfaces in Euclidean 3-space via the Gauss map, which is harmonic with respect to the metric induced by the second fundamental form. We give a method for constructing germs of Lorentzian-harmonic maps from their $k$-jets and use this construction to study the singularities of such maps. We also show how to construct pseudospherical surfaces with prescribed singularities using loop groups. We study the singularities of pseudospherical surfaces and obtain their bifurcations in generic 1-parameter families of such surfaces.

Example of Bianchi Transformation of Kuen’s Surface

The work is devoted to the study of the Bianchi transformation for surfaces of constant negative Gaussian curvature. The surfaces of rotation of constant negative Gaussian curvature are the Minding top, the Minding coil, and the pseudosphere (Beltrami surface). Surfaces of constant negative Gaussian curvature also include Kuen’s surface and the Dini’s surface. Studying the surfaces of constant negative Gaussian curvature (pseudospherical surfaces) is of great importance for the interpretation of Lobachevsky planimetry. Geometric characteristics of pseudospherical surfaces are found to be related to the theory of networks, the theory of solitons, nonlinear differential equations, and sin-Gordon equations. 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 transformation for the Kuen’s surface is constructed using a mathematical software package.

Преобразование Бианки волчка Миндинга

The work is devoted to the study of the Bianchi transform for surfac­es of revolution 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). 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 Minding top is constructed. Using a mathematical package, Minding's top and its Bianchi transform are constructed.

Full-parameter discrete Painlevé systems from non-translational Cremona isometries

Since the classification of discrete Painlevé equations in terms of rational surfaces, there has been much interest in the range of integrable equations arising from each of the 22 surface types in Sakai’s list (Sakai 2001 Commun. Math. Phys. 220 165–229). For all but the most degenerate type in the list, the surfaces come in families which admit affine Weyl groups of symmetries, translation elements of which define discrete Painlevé equations with the same number of parameters as their family of surfaces. While non-translation elements of the symmetry group have been observed to correspond to discrete systems of Painlevé-type through projective reduction, the resulting equations have fewer than the maximal number of free parameters corresponding to their surface type. We show that equations with the full number of free parameters can be constructed from non-translation elements of infinite order in the symmetry group, constructing several examples and demonstrating their integrability. This is prompted by the study of a previously proposed discrete Painlevé equation related to a special class of discrete analogues of surfaces of constant negative Gaussian curvature (Hoffmann 1999 Oxford Lect. Ser. Math. Appl. 16 83–96). We obtain a full-parameter generalisation of this equation from the Cremona action of a non-translation element of the extended affine Weyl group on a family of generic -surfaces.

The intersection of a hyperplane with a lightcone in the Minkowski spacetime

Klainerman, Luk and Rodnianski derived an anisotropic criterion for formation of trapped surfaces in vacuum, extending the original trapped surface formation theorem of Christodoulou. The effort to understand their result led us to study the intersection of a hyperplane with a lightcone in the Minkowski spacetime. For the intrinsic geometry of the intersection, depending on the hyperplane being spacelike, null or timelike, it has the constant positive, zero or negative Gaussian curvature. For the extrinsic geometry of the intersection, we find that it is a noncompact marginal trapped surface when the hyperplane is null. In this case, we find a geometric interpretation of the Green's function of the Laplacian on the standard sphere. In the end, we contribute a clearer understanding of the anisotropic criterion for formation of trapped surfaces in vacuum.

Constant negative Gaussian curvature tori and their singularities

We construct constant negative Gaussian curvature tori with one family of planar curvature lines in Euclidean 3-space. We show that these tori are wave fronts. The singularities of these tori are studied.

Lobachevsky crystallography made real through carbon pseudospheres

We realize Lobachevsky geometry in a simulation lab, by producing a carbon-based energetically stable molecular structure, arranged in the shape of a Beltrami pseudosphere. We find that this structure: (i) corresponds to a non-Euclidean crystallographic group, namely a loxodromic subgroup of ; (ii) has an unavoidable singular boundary, that we fully take into account. Our approach, substantiated by extensive numerical simulations of Beltrami pseudospheres of different size, might be applied to other surfaces of constant negative Gaussian curvature, and points to a general procedure to generate them. Our results also pave the way to test certain scenarios of the physics of curved spacetimes owing to graphene’s unique properties.

Hanbury Brown and Twiss measurements in curved space

Hanbury Brown and Twiss measurements on speckle patterns propagating along curved surfaces improve understanding of spatial coherence. When Hanbury Brown and Twiss (HBT) proposed their technique of intensity correlation measurements1,2,3 to examine the angular size of stars in the visible range, they challenged the common conception of quantum mechanics and kicked off a discussion that led to the establishment of quantum optics4,5,6. In this Letter we revisit this fundamental technique and study its implications in the presence of space curvature. To this end we theoretically and experimentally investigate the evolution of speckle patterns propagating along two-dimensional surfaces of constant positive and negative Gaussian curvature, defying the notion that light always gains spatial coherence during free-space propagation. We also discuss the measurability of the traversed space's curvature utilizing HBT from an inhabitant's point of view. Through their symmetry, surfaces with constant Gaussian curvature act as analogue models for universes possessing non-vanishing cosmological constants.

On Bäcklund and Ribaucour transformations for surfaces with constant negative curvature

Given a surface in the Euclidean 3-space with constant negative Gaussian curvature, the composition of Backlund transformations generates a 4-parameter family of surfaces with the same curvature. Since Ribaucour transformations of such a surface also provides a 4-parameter family of surfaces of the same type, one has the following natural question. Are these two methods equivalent? The answer is negative in general. Necessary and sufficient conditions for the surfaces given by the two procedures to be congruent are obtained. An explicit example is given of a composition of Backlund transformations which is not a Ribaucour transformation. The analytic interpretation of this result provides a method of producing solutions of the sine-Gordon equation, distinct from those obtained by Backlund transformations. New exact solutions are given and some of the corresponding surfaces are visualized.

Viscous fluid fingering on a negatively curved surface.

Viscous fingering formation in flat Hele-Shaw cells is a classical and widely studied fluid mechanical problem. We examine the development of viscous fluid fingering on a two-dimensional surface of constant negative Gaussian curvature, the hyperbolic plane H(2). A perturbative mode-coupling formalism is applied to study the influence of the negative surface curvature on the two most important pattern formation mechanisms of the system: fingertip splitting and finger competition. We also report on a time-dependent control strategy placed on the injection rate, which is able to minimize viscous fingering growth on H(2).

Minding's Theorem for Low Degrees of Differentiability

We prove Minding's Theorem for $C^2$-immersions with constant negative Gaussian curvature. As a Corollary we also prove Minding's Theorem for $C^{1M}$-immersions in the sense of [1].

Graphene: QFT in curved spacetimes close to experiments

A recently proposed step-by-step procedure, to merge the low-energy physics of the π-bonds electrons of graphene, and quantum field theory on curved spacetimes, is recalled. The last step there is the proposal of an experiment to test a Hawking-Unruh effect, emerging from the model, that manifests itself as an exact (within the model) prediction for the electronic local density of states, in the ideal case of the graphene membrane shaped as a Beltrami pseudosphere. A discussion about one particular attempt to experimentally test the model on molecular graphene is presented, and it is taken as an excuse to solve some basic issues that will help future experiments. In particular, it is stated that the effect should be visible on generic surfaces of constant negative Gaussian curvature, that are infinite in number.

Shape transitions in hyperbolic non-Euclidean plates

A non-Euclidean plate is a thin elastic object whose intrinsic geometry is not flat and hence has residual stresses arising from being embedded in three dimensional space. Recently, there has been interest in using localized swelling to induce residual stresses that shape flat objects into desired three dimensional structures. A fundamental question is whether we can use the mathematical theory of non-Euclidean plates to deduce the three dimensional configuration of the swelling sheet given the exact knowledge of the imposed geometry. We present and summarize the results of recent mathematical studies on non-Euclidean plates with imposed constant negative Gaussian curvature in both annular and disc geometries. We show in the Foppl–von Karman approximation to the elastic energy there are only two types of global minimizers – flat and saddle shaped deformations – with localized regions of stretching near the boundary of the domain. We also show that there exist n-wave local minimizers that closely resemble experimental observations and have additional regions of stretching near lines of inflection. Furthermore, in the Kirchhoff approximation to the elastic energy, we show that there exists exact isometric immersions with periodic profiles. The number of waves in these configurations is set by the condition that the bending energy remains finite and grows approximately exponentially with the radius of the annulus. For large radii, these shape are energetically favorable over saddle shapes and could explain why wavy shapes are selected in crochet models of the hyperbolic plane. The predicted morphologies however differ from what is observed in experiments on hydrogel disks highlighting the need for further theoretical studies.

Bounds of percolation thresholds on hyperbolic lattices

We analytically study bond percolation on hyperbolic lattices obtained by tiling a hyperbolic plane with constant negative Gaussian curvature. The quantity of our main concern is p(c2), the value of occupation probability where a unique unbounded cluster begins to emerge. By applying the substitution method to known bounds of the order-5 pentagonal tiling, we show that p(c2) ≥ 0.382508 for the order-5 square tiling, p(c2) ≥ 0.472043 for its dual, and p(c2)≥ 0.275768 for the order-5-4 rhombille tiling.