Non-constant Curvature Research Articles

Natural and conjugate mates of Frenet curves in three-dimensional Lie group

In this study, we introduce the natural mate and conjugate mate of a Frenet curve in a three dimensional Lie group $ \mathbb{G} $ with bi-invariant metric. Also, we give some relationships between a Frenet curve and its natural mate or its conjugate mate in $ \mathbb{G} $. Especially, we obtain some results for the natural mate and the conjugate mate of a Frenet curve in $ \mathbb{G} $ when the Frenet curve is a general helix, a slant helix, a spherical curve, a rectifying curve, a Salkowski (constant curvature and non-constant torsion), anti-Salkowski (non-constant curvature and constant torsion), Bertrand curve. Finally, we give nice graphics with numeric solution in Euclidean 3-space as a commutative Lie group.

Nanostructure regularity in white beetle scales for stability and strong optical scattering [Invited

Cyphochilus white beetle scales exhibit exceptionally strong light scattering power that originates from their regular random fibrillar network nanostructure. The structure is believed to be formed by late-stage spinodal decomposition in a lipid membrane system. However, the structure is characterized by nonconstant mean curvatures and appreciable anisotropy, which are not expected from late-stage spinodal decomposition, so that the surface free energy is not minimized. Nevertheless, a high degree of regularity represented by the relatively uniform fibril dimensions and smooth fibril surfaces in the structure may result from a process similar to spinodal decomposition. In this study, we investigate the role of regularity in the Cyphochilus white beetle scale structure in realizing strong light scattering. Irregularity is computationally introduced into the structure in a systematic fashion such that its anisotropy is preserved and its surface area is kept constant. Calculations show that optical scattering power decreases as irregularity increases with a high sensitivity. This effect happens because, remarkably, irregularity on a scale much smaller than the wavelength destroys anisotropy in optical diffusion. Thus, the result shows that the in vivo process in Cyphochilus white beetle scales utilizes structural regularity and anisotropy to achieve strong light scattering at a tolerable surface free energy. In typical fabrication of random media, irregularity and multiple length scales typically increase surface area, so that durability of the nanostructures may be negatively affected. Our study indicates that regularity in anisotropic random nanostructures can achieve strong light scattering with a moderate surface free energy.

Comment on ,,On the integrability of 2D Hamiltonian systems with variable Gaussian curvature\u201d by A. A. Elmandouh

In the paper [1], the author formulates in Theorem 2 necessary conditions for integrability of a certain class of Hamiltonian systems with non-constant Gaussian curvature, which depends on local coordinates. We give a counterexample to show that this theorem is not correct in general. This contradiction is explained in some extent. However, the main result of this note is our theorem that gives new simple and easy to check necessary conditions to integrability of the system considered in [1]. We present several examples, which show that the obtained conditions are effective. Moreover, we justify that our criterion can be extended to wider class of systems, which are given by non-meromorphic Hamiltonian functions.

A large displacement model for superelastic material side-notched tube instruments

Side-notched tube structures made of superelastic material are increasingly being studied in surgical robotics as they can easily be integrated into surgical instruments and offer the surgeon enhanced dexterity. Their simple design and ability to achieve large bending angles and small bending radii make them particularly suited for use in constrained workspace surgery. Up to now, however, no model has been able to accurately predict the behavior of such types of structures for large bending angles. This study, therefore, proposes a novel approach to model large displacements of side-notched tubes taking into account the superelasticity of the material, the non-constant curvature of the side-notched structure, as well as the friction induced by the actuating wire. More specifically, an equivalent, strain dependent, linear elastic modulus is used to model the superelastic behavior of the instrument’s material. The deformation of each separate section of the side-notched tube is calculated successively to capture the non-constant curvature of the NiTi backbone. The capstan equation is used to model cable friction. The model was tested on four samples of 2.3 mm diameter and was able to predict the bending angle of the side-notched tubes with a root mean square error (RMSE) of as low as 5.4∘ (3.1%) and the position of each notch with an RMSE of as low as 0.33 mm (2.5%) across its entire bending range (0∘ to 180∘). The model was demonstrated for both tension and relaxation of the actuating wire.

Thin shells in F(R) gravity with non-constant scalar curvature

We introduce two classes of spherically symmetric spacetimes having a thin shell of matter, in non-quadratic F(R) theories of gravity with non-constant scalar curvature R. In the first, the thin shell joins an inner region with an outer one, while in the second it corresponds to the throat of a wormhole. In both scenarios, we analyze the stability of the static configurations under radial perturbations. As particular examples in spacetimes with a cosmological constant, we present charged thin shells surrounding a non-charged black hole and charged thin-shell wormholes. We show that in both cases stable solutions are possible for suitable values of the parameters.

Totally umbilical hypersurfaces of Spinc manifolds carrying special spinor fields

Under some dimension restrictions, we prove that totally umbilical hypersurfaces of Spin[Formula: see text] manifolds carrying a parallel, real or imaginary Killing spinor are of constant mean curvature. This extends to the Spin[Formula: see text] case the result of Kowalski stating that, every totally umbilical hypersurface of an Einstein manifold of dimension greater or equal to [Formula: see text] is of constant mean curvature. As an application, we prove that there are no extrinsic hypersheres in complete Riemannian [Formula: see text] manifolds of non-constant sectional curvature carrying a parallel, Killing or imaginary Killing spinor.

Thermodynamical correspondence of f(R) gravity in the Jordan and Einstein frames

We study the thermodynamical aspects of [Formula: see text] gravity in the Jordan and the Einstein frame, and we investigate the corresponding equivalence of the thermodynamical quantities in the two frames. We examine static spherically symmetric black hole solutions with constant Ricci scalar curvature [Formula: see text], and as we demonstrate, the thermodynamical quantities in the two frames are equivalent. However, for the case of black holes with nonconstant scalar curvature [Formula: see text], the thermodynamical equivalence of the two frames is no longer valid. In addition, we extend our study to investigate cosmological solutions with a homogeneous and isotropic background. In particular, we find that the power-law cosmology case provides an accidentally thermodynamical equivalence of the two frames. However, for nontrivial cosmology, we examine a novel exponential ultraviolet [Formula: see text] gravity. This confirms that the thermodynamical quantities in both frames are not equivalent. In conclusion, although [Formula: see text] gravity and its corresponding scalar-tensor theory are mathematically equivalent, at least for conformal invariant quantities, the two frames are not thermodynamically equivalent at a quantitative level, in terms of several physical quantities.

Soft Robotic Gripper With Compliant Cell Stacks for Industrial Part Handling

Robot object grasping and handling requires accurate grasp pose estimation and gripper/end-effector design, tailored to individual objects. When object shape is unknown, cannot be estimated, or is highly complex, parallel grippers can provide insufficient grip. Compliant grippers can circumvent these issues through the use of soft or flexible materials that adapt to the shape of the object. This letter proposes a 3D printable soft gripper design for handling complex shapes. The compliant properties of the gripper enable contour conformation, yet offer tunable mechanical properties (i.e., directional stiffness). Objects that have complex shape, such as non-constant curvature, convex and/or concave shape can be grasped blind (i.e., without grasp pose estimation). The motivation behind the gripper design is handling of industrial parts, such as jet and Diesel engine components. (Dis)assembly, cleaning and inspection of such engines is a complex, manual task that can benefit from (semi-)automated robotic handling. The complex shape of each component, however, limits where and how it can be grasped. The proposed soft gripper design is tunable by compliant cell stacks that deform to the shape of the handled object. Individual compliant cells and cell stacks are characterized and a detailed experimental analysis of more than 600 grasps with seven different industrial parts evaluates the approach.

Curvature-induced skyrmion mass

We investigate the propagation of magnetic skyrmions on elastically deformable geometries by employing imaginary time quantum field theory methods. We demonstrate that the Euclidean action of the problem carries information of the elements of the surface space metric, and develop a description of the skyrmion dynamics in terms of a set of collective coordinates. We reveal that novel curvature-driven effects emerge in geometries with non-constant curvature, which explicitly break the translational invariance of flat space. In particular, for a skyrmion stabilized by a curvilinear defect, an inertia term and a pinning potential are generated by the varying curvature, while both of these terms vanish in the flat-space limit.

Memory effects in Kundt wave spacetimes

Memory effects in the exact Kundt wave spacetimes are shown to arise in the behaviour of geodesics in such spacetimes. The types of Kundt spacetimes we consider here are direct products of the form H2×M(1,1) and S2×M(1,1). Both geometries have constant scalar curvature. We consider a scenario in which initial velocities of the transverse geodesic coordinates are set to zero (before the arrival of the pulse) in a spacetime with non-vanishing background curvature. We look for changes in the separation between pairs of geodesics caused by the pulse. Any relative change observed in the position and velocity profiles of geodesics, after the burst, can be solely attributed to the wave (hence, a memory effect). For constant negative curvature, we find there is permanent change in the separation of geodesics after the pulse has departed. Thus, there is displacement memory, though no velocity memory is found. In the case of constant positive scalar curvature (Plebański–Hacyan spacetimes), we find both displacement and velocity memory along one direction. In the other direction, a new kind of memory (which we term as frequency memory effect) is observed where the separation between the geodesics shows periodic oscillations once the pulse has left. We also carry out similar analyses for spacetimes with a non-constant scalar curvature, which may be positive or negative. The results here seem to qualitatively agree with those for constant scalar curvature, thereby suggesting a link between the nature of memory and curvature.

ManifoldNet: A Deep Neural Network for Manifold-Valued Data With Applications.

Geometric deep learning is a relatively nascent field that has attracted significant attention in the past few years. This is partly due to the availability of data acquired from non-euclidean domains or features extracted from euclidean-space data that reside on smooth manifolds. For instance, pose data commonly encountered in computer vision reside in Lie groups, while covariance matrices that are ubiquitous in many fields and diffusion tensors encountered in medical imaging domain reside on the manifold of symmetric positive definite matrices. Much of this data is naturally represented as a grid of manifold-valued data. In this paper we present a novel theoretical framework for developing deep neural networks to cope with these grids of manifold-valued data inputs. We also present a novel architecture to realize this theory and call it the ManifoldNet. Analogous to vector spaces where convolutions are equivalent to computing weighted sums, manifold-valued data 'convolutions' can be defined using the weighted Fréchet Mean ([Formula: see text]). (This requires endowing the manifold with a Riemannian structure if it did not already come with one.) The hidden layers of ManifoldNet compute [Formula: see text]s of their inputs, where the weights are to be learnt. This means the data remain manifold-valued as they propagate through the hidden layers. To reduce computational complexity, we present a provably convergent recursive algorithm for computing the [Formula: see text]. Further, we prove that on non-constant sectional curvature manifolds, each [Formula: see text] layer is a contraction mapping and provide constructive evidence for its non-collapsibility when stacked in layers. This captures the two fundamental properties of deep network layers. Analogous to the equivariance of convolution in euclidean space to translations, we prove that the [Formula: see text] is equivariant to the action of the group of isometries admitted by the Riemannian manifold on which the data reside. To showcase the performance of ManifoldNet, we present several experiments using both computer vision and medical imaging data sets.

Spectrally positive Bakry-Émery Ricci curvature on graphs

We investigate analytic and geometric implications of non-constant Ricci curvature bounds. We prove a Lichnerowicz eigenvalue estimate and finiteness of the fundamental group assuming that L+2Ric is a positive operator where L is the graph Laplacian. Assuming that the negative part of the Ricci curvature is small in Kato sense, we prove diameter bounds, elliptic Harnack inequality and Buser inequality. This article seems to be the first one establishing these results while allowing for some negative curvature.

Single-point gradient blow-up on the boundary for diffusive Hamilton-Jacobi equation in domains with non-constant curvature

We consider the diffusive Hamilton-Jacobi equation ut−Δu=|∇u|p in a bounded planar domain with zero Dirichlet boundary condition. It is known that, for p>2, the solutions to this problem can exhibit gradient blow-up (GBU) at the boundary. In this paper we study the possibility of the GBU set being reduced to a single point. In a previous work [Y.-X. Li, Ph. Souplet, 2009], it was shown that single point GBU solutions can be constructed in very particular domains, i.e. locally flat domains and disks. Here, we prove the existence of single point GBU solutions in a large class of domains, for which the curvature of the boundary may be nonconstant near the GBU point.Our strategy is to use a boundary-fitted curvilinear coordinate system, combined with suitable auxiliary functions and appropriate monotonicity properties of the solution. The derivation and analysis of the parabolic equations satisfied by the auxiliary functions necessitate long and technical calculations involving boundary-fitted coordinates.

Control Oriented Modeling of Soft Robots: The Polynomial Curvature Case

The complex nature of soft robot dynamics calls for the development of models specifically tailored on the control application. In this letter, we take a first step in this direction by proposing a dynamic model for slender soft robots, taking into account the fully infinite-dimensional dynamical structure of the system. We also contextually introduce a strategy to approximate this model at any level of detail through a finite dimensional system. First, we analyze the main mathematical properties of this model in the case of lightweight and non lightweight soft robots. Then, we prove that using the constant term of curvature as control output produces a minimum phase system, in this way providing the theoretical support that existing curvature control techniques lack, and at the same time opening up to the use of advanced nonlinear control techniques. Finally, we propose a new controller, i.e., the PD-poly, which exploits information on high order deformations, to achieve zero steady state regulation error in presence of gravity and generic nonconstant curvature conditions.

Convexity Properties of Harmonic Functions on Parameterized Families of Hypersurfaces

It is known that the $$L^{2}$$ -norms of a harmonic function over spheres satisfy some convexity inequality strongly linked to the Almgren’s frequency function. We examine the $$L^{2}$$ -norms of harmonic functions over a wide class of evolving hypersurfaces. More precisely, we consider compact level sets of smooth regular functions and obtain a differential inequality for the $$L^{2}$$ -norms of harmonic functions over these hypersurfaces. To illustrate our result, we consider ellipses with constant eccentricity and growing tori in $${\mathbf {R}}^3.$$ Moreover, we give a new proof of the convexity result for harmonic functions on a Riemannian manifold when integrating over spheres. The inequality we obtain for the case of positively curved Riemannian manifolds with non-constant curvature is slightly better than the one previously known.

Compact quotients of Cahen-Wallach spaces

Indecomposable symmetric Lorentzian manifolds of non-constant curvature are called Cahen-Wallach spaces. Their isometry classes are described by continuous families of real parameters. We derive necessary and sufficient conditions for the existence of compact quotients of Cahen-Wallach spaces in terms of these parameters.

Vortex crystals on the surface of a torus

Vortex crystals are equilibrium states of point vortices whose relative configuration is unchanged throughout the evolution. They are examples of stationary point configurations subject to a logarithmic particle interaction energy, which give rise to phenomenological models of pattern formations in incompressible fluids, superconductors, superfluids and Bose–Einstein condensates. In this paper, we consider vortex crystals rotating at a constant speed in the latitudinal direction on the surface of a torus. The problem of finding vortex crystals is formulated as a linear null equation A Γ = 0 for a non-normal matrix A whose entities are derived from the locations of point vortices, and a vector Γ consisting of the strengths of point vortices and the latitudinal speed of rotation. Point configurations of vortex crystals are obtained numerically through the singular value decomposition by prescribing their locations and/or by moving them randomly so that the matrix A becomes rank deficient. Their strengths are taken from the null space corresponding to the zero singular values. The toroidal surface has a non-constant curvature and a handle structure, which are geometrically different from the plane and the spherical surface where vortex crystals have been constructed in the preceding studies. We find new vortex crystals that are associated with these toroidal geometry: (i) a polygonal arrangement of point vortices around the line of longitude; (ii) multiple latitudinal polygonal ring configurations of point vortices that are evenly arranged around the handle; and (iii) point configurations along helical curves corresponding to the fundamental group of the toroidal surface. We observe the strengths of point vortices and the behaviour of their distribution as the number of point vortices gets larger. Their linear stability is also examined. This article is part of the theme issue ‘Topological and geometrical aspects of mass and vortex dynamics’.

Eta-Ricci solitons on LP-Sasakian manifolds

We consider η-Ricci solitons on Lorentzian para-Sasakian manifolds with Codazzi type of the Ricci tensor. Then we study η-Ricci solitons on ϕ-conformally semi-symmetric, ϕ-Ricci symmetric, and conformally Ricci semi-symmetric Lorentzian para-Sasakian manifolds. Finally, we construct an example of a three dimensional Lorentzian para-Sasakian manifold which admits η-Ricci solitons with non-constant scalar curvature.

Petrov type-N solution for the null-surface formulation in $$2+1$$ dimensions

A nontrivial solution of the field equations of the null-surface formulation in $$3+1$$ dimensions has not, as yet, been found and, even in $$2+1$$ dimensions, the field equations are extremely difficult to solve. Only two nontrivial solutions in $$2+1$$ dimensions have previously been found and this work presents a third (exact) solution. It differs markedly from the two earlier solutions not only in its functional nature (parametric, as opposed to explicit or implicit) but also in its Petrov type and the physical nature of the source term. The solution is expressed in terms of elementary functions and has positive non-constant scalar curvature. The spacetime has a curvature singularity and the source of the gravitational field can be interpreted as a minimally coupled scalar field with a Liouville potential. The strong energy condition is violated but all of the other energy conditions hold. There is a congruence of null geodesic Killing vectors with the expansion, shear, and vorticity scalars all being zero.

Construction of vacuum initial data by the conformally covariant split system

Using the implicit function theorem, we prove existence of solutions of the so-called conformally covariant split system on compact 3-dimensional Riemannian manifolds. They give rise to non-constant mean curvature (non-CMC) vacuum initial data for the Einstein equations. We investigate the conformally covariant split system defined on compact manifolds with or without boundaries. In the former case, the boundary corresponds to an apparent horizon in the constructed initial data. The case with a cosmological constant is then considered separately. Finally, to demonstrate the applicability of the conformal covariant split system in numerical studies, we provide numerical examples of solutions on manifolds and .