Surfaces Of Gaussian Curvature Research Articles

ON THE GAUSSIAN CURVATURE FUNCTIONAL IN THE CLASS OF SURFACES OF POSITIVE GAUSSIAN CURVATURE

The paper establishes the form of the Gaussian curvature functional defined on the class of infinitely differentiable horizontal surfaces of positive Gaussian curvature. With respect to admissible surfaces, it is assumed that they admit a global semi-geodetic parametrization. The paper proves that the first variation of the functional on the class of variations of admissible surfaces admitting connections between the coefficients of the first quadratic form and their geodesic lines similar to the axisymmetric case is determined by the Gaussian curvature of the varied surface. The considerations of the type are closely connected with the problems of the study of the equilibrium forms having one of the main curvatures sufficiently small. In this case, the classical Laplace formula fails. Thus, there appear a necessity to take into account more subtle processes leading to the adequate description of the equilibrium state of the two-phased system. In particular, it is quite natural to introduce into the study an intermediate layer consisting of the molecules of the two different phases, one of the Maxwell’s ideas. The calculations of the work spended by the pressure forces for the formation of intermediate layer leads us to the necessity to introduce Gauss functional into consideration. Linear combination of mean curvature and gaussian curvature functionals gives possibility to construct variational solution of generalized Laplace equation.

Schwarz lemma for conformal parametrization of minimal graphs in [formula omitted]

We prove Schwarz-type lemma results for Weierstrass parameterization of the minimal disk in the Riemannian manifold M×R, where M is a Riemannian surface of non-positive Gaussian curvature. The estimate is sharp, and the equality is attained if and only if the ϱ-harmonic mapping that produces the parameterization is conformal and the metric is a Euclidean metric. If the area of the minimal surface is equal to the area of the disk, then the parametrization is a contraction w.r.t. induced metric and hyperbolic metric respectively.

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.

Rotating square tessellations enabled stretchable and adaptive curved display

Curved displays can adjust their shape to accommodate different objects and are used in electronics and decorative lighting. Due to the immutable pixel spacing, existing commercial curved displays are flexible but not compatible with undevelopable surfaces. Inspired by kirigami and auxetic structures, we propose an approach that combines luminescent elements and rotating square tessellations to create a stretchable, arbitrary curve adaptive display. We connect square islands by vertical interconnects to relieve the stress concentration and provide extra deformation patterns. The vertical interconnects are patterned on a flexible printed circuit board (FPCB) using laser cutting and folded up via specially designed molds. Further, the freed-up space by folded interconnects allows the structure to be compressed. A prototype stretchable display is demonstrated that it can maintain electrical performance under biaxial strain and adapt to different Gaussian curvature surfaces, including cylindrical, spherical, saddle and arbitrary surfaces. Theoretical models and finite element calculations are established to describe the tensile behavior of the structures under different boundary conditions and agree with the experimental results. This proposed technology paves a feasible solution of mass production of adaptive curved displays and sets the trend for the next-generation display.

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.

Numerical and experimental investigation of a variable-curvature shell class I flextensional transducer.

Class I flextensional transducers cannot form broadband characteristics because of the deep valley between their first- and second-order response curves. This valley arises from the sound pressure cancellation produced by the vibration in different areas of the shell. This paper examines the cause of the deep valley by analyzing the three-dimensional equivalent radiated sound field of the transducer. By solving the three-dimensional vibration equation of the shell of revolution and reducing its dimensionality, it can be concluded that its vibration state is related to the positive and negative curvatures in both main directions. Therefore, a variable-curvature shell class I flextensional transducer is proposed, and the second-order vibration of the transducer is changed by the variable-curvature structure composed of positive and negative Gaussian curvature surfaces. The proposed transducer avoids sound pressure cancellation and achieves broadband characteristics. After optimization, a transducer prototype is fabricated, and a pool test is conducted. The test results show that the transducer couples the first three vibration modes, attaining broadband characteristics with an in-band fluctuation of less than 8 dB and a bandwidth of more than 5.4 kHz in the axial and circumferential directions.

ІНФІНІТЕЗИМАЛЬНІ ДЕФОРМАЦІЇ ПОВЕРХОНЬ ІЗ ЗАДАНОЮ ЗМІНОЮ ТЕНЗОРА РІЧЧІ

In three-dimensional Euclidean space, we study the problem of the existence of an infinitesimal first-order deformation of single-connected regular surfaces with a predetermined change in the Ricci tensor. It is shown that for surfaces of nonzero Gaussian curvature, this problem is reduced to the study and solution of a system of seven equations (including differential equations) with respect to seven unknown functions, each solution of which determines a vector field that is a univariate function (with an accuracy of a constant vector) and can be interpreted as a moment-free stress state of equilibrium of a loaded shell. For regular surfaces of non-zero Gaussian and mean curvatures, the problem is reduced to finding solutions to one second-order partial differential equation with respect to two unknown functions. Given one of these functions, the resulting equation will in general be a nonhomogeneous second-order partial differential equation (nonhomogeneous Weingarten differential equation). It is proved that any regular surface of positive Gaussian and non-zero mean curvature admits an infinitesimal first-order deformation with a given change in the Ricci tensor in a sufficiently small region. In this case, the tensor fields will be represented by an arbitrary and predefined regular function. By considering the Neumann problem, it is shown that a single-connected regular surface of elliptic type of positive Gaussian and negative mean curvature with a regular boundary under a certain boundary condition admits, in general, an infinitesimal first-order deformation with a predetermined change in the Ricci tensor. In this case, the tensor fields will be determined uniquely. For surfaces of negative Gaussian and non-zero mean curvature, the resulting inhomogeneous partial differential equation with second-order partial differentials will be of hyperbolic type with known coefficients and right-hand side. The Darboux problem is considered for this equation. It is proved that any regular surface of negative Gaussian and non-zero mean curvature admits an infinitesimal first-order deformation with a given change in the Ricci tensor. Tensor fields are expressed through a given function of two variables and through two arbitrary regular functions of one variable. Keywords: infinitesimal deformation, Ricci tensor, tensor fields, Gaussian curvature, mean curvature.

Rotational surfaces in a 3-dimensional normed space

We study rotational surfaces with constant Minkowski Gaussian curvature and rotational surfaces with constant Minkowski mean curvature in a 3-dimensional normed space with rotationally symmetric norm. We have a generalization of the catenoid, pseudo-sphere and Delaunay surfaces.

‘Classical’ coherent state generated by curved surface

Analogous coherent states are deduced from classical optical fields on curved surface in this paper. The Gaussian laser beam, as a fundamental mode, cannot be adequately simulated by coherent states due to their inherent diffraction in flat space. But it differs when propagating on a surface with the uniform curvature, the constant Gaussian curvature surface (CGCS). By generalizing the method of Feynman path integral, an equivalent coherent states solution is demonstrated to describe the beam propagation. The temporal evolution of the Schrödinger equation is analogously translated into a spatial transmission in this derivation, and we obtain the expression of quantized momentum transmitted on curved surface, which is proportional to the square root of the Gaussian curvature K. In addition to that, a physical picture of beam propagation that is identical to the squeezed state is also built. We hope this research can offer a new view on the quantum field in curved space.

Decouplings for surfaces of zero curvature

We extend the l2(Lp) decoupling theorem of Bourgain-Demeter to the full class of developable surfaces in R3. This completes the l2 decoupling theory of the zero Gaussian curvature surfaces that lack planar (or umbilic) points. Of central interest to our study is the tangent surface associated to the moment curve.

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.

Propagation properties of finite Airy beams on curved surfaces.

Airy beams have provided exciting inspiration in the field of optical communication, particle manipulation, and imaging. We investigate the propagation properties of the exponential truncation Airy beams (ETABs) on constant Gaussian curvature surfaces (CGCSs) in this paper. The analytical expression of the electric field of ETABs propagating on the CGCSs is derived. It shows that the equivalent periodical accelerations of the trajectories of ETABs on the curved surface are always larger than the constant one on the flat surface because the CGCSs have a strong focusing ability. For the same reason, the non-diffraction propagation of ETABs is found when the focusing ability of the CGCSs is strong enough. Moreover, we investigate the self-healing length of ETABs on CGCSs and explore that the ability of self-healing is related to the geometry of CGCSs besides the width of the block and the size of the beam. The self-healing length gets larger with the increase of radius of CGCSs and finally consists with that on the flat surface. These propagation characteristics are different from those in the flat space and are useful for the future applications of ETABs in particle manipulation on waveguides, light-sheet fluorescence microscopy, curved nanophotonics, and so on.

Shape Prediction of the Sheet in Continuous Roll Forming Based on the Analysis of Exit Velocity.

Continuous roll forming (CRF) is a new technology that combines continuous forming and multi-point forming to produce three-dimensional (3D) curved surfaces. Compared with other methods, the equipment of CRF is very simple, including only a pair of bendable work rolls and the corresponding shape adjustment and support assembly. By controlling the bending shapes of the upper and lower rolls and the size of the roll gap during forming, double curvature surfaces with different shapes can be produced. In this paper, a simplified expression of the exit velocity of the sheet is provided, and the formulas for the calculation of the longitudinal curvature radius are further derived. The reason for the discrepancy between the actual and predicted values of the longitudinal radius is deeply discussed from the perspective of the distribution of the exit velocity. By using the response surface methodology, the effects of the maximum compression ratio, the sheet width, the sheet thickness, and the transverse curvature radius on the longitudinal curvature radius are analyzed. Meanwhile, the correction coefficients of the predicted formulas for the positive and negative Gaussian curvature surfaces are obtained as 1.138 and 0.905, respectively. The validity and practicability of the modified formulas are verified by numerical simulations and forming experiments.

The hollow Gaussian beam propagation on curved surface based on matrix optics method

In this paper, based on the Euler–Lagrange equation, an ABCD matrix is constructed out to study the paraxial transmission of light on a constant Gaussian curvature surface (CGCS), which is the first time to our knowledge. Then, by using the method of matrix optics, we extend the CGCS matrix to a general transfer matrix which is suitable for a gently varying curvature. As a beam propagation example, based on the Collins integral and the derived ABCD matrix elements, an analytical propagation formula for the hollow Gaussian beams (HGBs) on the CGCS is deduced. The propagation characteristics of HGBs on a CGCS are illustrated graphically in detail, mainly including the change of dark spot size and splitting rays. Besides its propagation periodicity and diffraction properties, a criterion for convergence and divergence of the spot size is proposed. The area of the dark region of the HGBs can easily be controlled by proper choice of the beam parameters and the shape of CGCS. In addition, we also study the special propagation properties of the hollow beam with fractional order. Compared with propagation characteristics of HGBs in flat space, these novel propagation characteristics of HGBs on curved surface may further expand the application range of hollow beams.

Hyperbolic Center of Mass for a System of Particles in a Two-Dimensional Space with Constant Negative Curvature: An Application to the Curved 2-Body Problem

In this article, a simple expression for the center of mass of a system of material points in a two-dimensional surface of Gaussian constant negative curvature is given. By using the basic techniques of geometry, we obtained an expression in intrinsic coordinates, and we showed how this extends the definition for the Euclidean case. The argument is constructive and serves to define the center of mass of a system of particles on the one-dimensional hyperbolic sphere LR1.

Discrete Gaussian Curvature Flow for Piecewise Constant Gaussian Curvature Surface

A method is presented for generating a discrete piecewise constant Gaussian curvature (CGC) surface. An energy functional is first formulated so that its stationary point is the linear Weingarten (LW) surface, which has a property such that the weighted sum of mean and Gaussian curvatures is constant. The CGC surface is obtained using the gradient derived from the first variation of a special type of the energy functional of the LW surface and updating the surface shape based on the Gaussian curvature flow. A filtering method is incorporated to prevent oscillation and divergence due to unstable property of the discretized Gaussian curvature flow. Two techniques are proposed to generate a discrete piecewise CGC surface with preassigned internal boundaries. The step length of Gaussian curvature flow is adjusted by introducing a line search algorithm to minimize the energy functional. The effectiveness of the proposed method is demonstrated through numerical examples of generating various shapes of CGC surfaces.

ДЕФОРМАЦІЇ ПОВЕРХОНЬ ЗІ СТАЦІОНАРНИМ ТЕНЗОРОМ РІЧЧІ

In this paper, we consider infinitesimal (n. m.) first-order deformations of single-connected regular surfaces in three-dimensional Euclidean space. The search for the vector field of this deformation is generally reduced to the study and solution of a system of four equations (among them there are differential equations) with respect to seven unknown functions. To avoid uncertainty, the following restriction is imposed on a given surface: the Ricci tensor is stored (mainly) on the surface. A mathematical model of the problem is created: a system of seven equations with respect to seven unknown functions. Its mechanical content is established. It is shown that each solution of the obtained system of equations will determine the field of displacement n. m. deformation of the first order of the surface of nonzero Gaussian curvature, which will be an unambiguous function (up to a constant vector). It is proved that each regular surface of nonzero Gaussian and mean curvatures allows first-order n. m. deformation with a stationary Ricci tensor. The tensor fields are found explicitly and depend on two functions, which are the solution of a linear inhomogeneous second-order differential equation with partial derivatives. The class of rigid surfaces in relation to the specified n. m. deformations. Assuming that one of the functions is predetermined, the obtained differential equation in the General case will be a inhomogeneous differential Weingarten equation, and an equation of elliptical type. The geometric and mechanical meaning of the function that is the solution of this equation is found. The following result was obtained: any surface of positive Gaussian and nonzero mean curvatures admits n. m of first-order deformation with a stationary Ricci tensor in the region of a rather small degree. Tensor fields will be represented by a predefined function and some arbitrary regular functions. Considering the Dirichlet problem, it is proved that the simply connected regular surface of a positive Gaussian and nonzero mean curvatures under a certain boundary condition admits a single first-order deformation with a stationary Ricci tensor. The strain tensors are uniquely defined.

Fullerene‐like structures of Cretaceous crinoids reveal topologically limited skeletal possibilities

AbstractThere are few cases where numbers or types of possible phenotypes are known, although vast state spaces have been postulated. Rarely applied in this context, graph theory and topology enable enumeration of possible phenotypes and evolutionary transitions. Here, we generate polyhedral calyx graphs for the Late Cretaceous, stemless crinoids Marsupites testudinarius and Uintacrinus socialis (Uintacrinoidea Zittel) revealing structural similarities to carbon fullerene and fulleroid molecules (respectively). The U. socialis calyx incorporates numerous plates (e.g. graph vertices |V| ≥ 197), which are small, light, low‐density and have four to eight sides. Therefore, the corresponding number of possible plate arrangements (number of polyhedral graphs) is large (≫1 × 1014). Graph vertices representing plates with sides >6 introduce negative Gaussian curvature (surface saddle points) and topological instability, increasing buckling risk. However, observed numbers of vertices for Uintacrinus do not allow more stable pentaradial configurations. In contrast, the Marsupites calyx dual graph has 17 faces that are pentagonal or hexagonal. Therefore, it is structurally identical to a carbon fullerene, specifically C30‐D5h. Corresponding graph restrictions result in constraint to only three structural options (fullerene structures C30‐C2v 1, C30‐C2v 2 and C30‐D5h). Further restriction to pentaradial symmetry allows only one possibility: the Marsupites phenotype. This robust, stable topology is consistent with adaptation to predation pressures of the Mesozoic marine revolution. Consequently, the most plausible evolutionary pathway between unitacrinoid phenotypes was a mixed heterochronic trade‐off to fewer, larger calyx plates. Therefore, topological limitations radically constrained uintacrinoid skeletal possibilities but thereby aided evolution of a novel adaptive phenotype.

Asymptotically Poincaré surfaces in quasi-Fuchsian manifolds

We introduce the notion of an asymptotically Poincaré family of surfaces in an end of a quasi-Fuchsian manifold. We show that any such family gives a foliation of an end by asymptotically parallel convex surfaces, and that the asymptotic behavior of the first and second fundamental forms determines the projective structure at infinity. As an application, we establish a conjecture of Labourie from [J. London Math. Soc. 45 (1992), pp. 549–565] regarding constant Gaussian curvature surfaces. We also derive consequences for constant mean curvature surfaces.

Charged fermion in two-dimensional curved spaces of constant Gaussian curvature with constant magnetic flux

We investigate the behavior of spin-1/2 particles (electron and positron) confined to the Gaussian curvature surfaces. For the non-negative Gaussian curvatures, we present the preliminary results in cylindrical and spherical cases. To be specific we use the deformed hyperbolic solutions to obtain eigenvalues of the Dirac equation in the presence of an axial gauge field. Our results demonstrate the quantized energy and eigenstates of fermion. The quantization of energy depends on the spin-orbit coupling and the Landau quantization. The imaginary energy is obtained from the negative Gaussian curvatures. It is interpreted as the quasi normal mode (QNM). The angular momentum of fermion is shifted by addition of the constant magnetic flux. The fermion behaves like boson when the flux is half-integer.