Non-constant Curvature Research Articles

Refining Euclidean Obfuscatory Nodes Helps: A Joint-Space Graph Learning Method for Graph Neural Networks.

Many graph neural networks (GNNs) are inapplicable when the graph structure representing the node relations is unavailable. Recent studies have shown that this problem can be effectively solved by jointly learning the graph structure and the parameters of GNNs. However, most of these methods learn graphs by using either a Euclidean or hyperbolic metric, which means that the space curvature is assumed to be either constant zero or constant negative. Graph embedding spaces usually have nonconstant curvatures, and thus, such an assumption may produce some obfuscatory nodes, which are improperly embedded and close to multiple categories. In this article, we propose a joint-space graph learning (JSGL) method for GNNs. JSGL learns a graph based on Euclidean embeddings and identifies Euclidean obfuscatory nodes. Then, the graph topology near the identified obfuscatory nodes is refined in hyperbolic space. We also present a theoretical justification of our method for identifying obfuscatory nodes and conduct a series of experiments to test the performance of JSGL. The results show that JSGL outperforms many baseline methods. To obtain more insights, we analyze potential reasons for this superior performance.

Higher-curvature gravity in AdS3 , holographic c -theorems, and black hole microstates

We construct higher-derivative gravity theories in three dimensions that admit holographic c-theorems and exhibit a unique maximally symmetric vacuum, at arbitrary order n in the curvature. We show that these theories exhibit special properties, the most salient ones being the decoupling of ghost modes around anti–de Sitter (AdS) space, the enhancement of symmetries at linearized level, and the existence of a one-parameter generalization of the Bañados-Teitelboim-Zanelli (BTZ) black hole that, while being asymptotically AdS, is not of constant curvature but rather exhibits a curvature singularity. For such black holes, we provide a holographic derivation of their thermodynamics. This gives a microscopic picture of black hole thermodynamics for nonsupersymmetric solutions, of nonconstant curvature in higher-derivative theories of arbitrary order in the curvature. Published by the American Physical Society 2024

Levi-flat real hypersurfaces in nonflat complex planes

Let M be a non-ruled weakly 2-Hopf real hypersurface in a nonflat complex plane whose mean curvature is invariant along the Reeb flow. In this paper, it is proved that M is Levi-flat if and only if M is locally congruent to a strongly 2-Hopf real hypersurface of a special type. As a corollary we present a class of non-ruled Levi-flat real hypersurfaces of dimension three with non-constant mean curvature.

Study about black hole solutions with nonconstant transversal curvature and its conserved charges in Lovelock gravity

Study about black hole solutions with nonconstant transversal curvature and its conserved charges in Lovelock gravity

Composite continuum robots: Accurate modeling and model reduction

Models of continuum robots often trade off accuracy and speed. The constant curvature assumption model can perform fast calculations but with poor accuracy. Static models can obtain good calculation accuracy but with slower computation speed. In this paper, a precise and efficient method for establishing a non-constant curvature kinematic model for composite continuum robots is proposed. First, an algorithm based on the Beam Constraint Model is proposed to establish the static model of the continuum robot. The situation in which friction changes gradually in continuum robots is considered in the static model. Then, the Kepler elliptic curve is used to simplify the static model to build a non-constant curvature kinematic model of the continuum robot. Then, complex inverse kinematics solutions are transformed into explicit sets of equations by mathematical transformation. Finally, simulations and experiments are designed to demonstrate the algorithm proposed in this paper. The simulation results show that the proposed algorithm is 1572 times faster than the Levenberg–Marquardt algorithm. The experimental results show that the average error of the static model is 0.065 mm, accounting for 0.13% of its length. The maximum error is 0.309 mm, accounting for 0.618% of its length. The positioning accuracy of the non-constant curvature model proposed in this paper is 2.02 times that of the constant curvature model. The results show that the simplified kinematic model algorithm based on the static model has high accuracy and computational efficiency. This research is crucial for the practical application of continuum robots in engineering.

The pluriclosed flow for T2-invariant Vaisman metrics on the Kodaira-Thurston surface

In this note we study T2-invariant pluriclosed metrics on the Kodaira-Thurston surface. We obtain a characterization of T2-invariant Vaisman metrics, and notice that the Kodaira-Thurston surface admits Vaisman metrics with non-constant scalar curvature. Then we study the behaviour of the Vaisman condition in relation to the pluriclosed flow. As a consequence, we show that if the initial metric on the Kodaira-Thurston surface is a T2-invariant Vaisman metric, then the pluriclosed flow preserves the Vaisman condition, extending to the non-constant scalar curvature case the previous result in [6].

Small spheres with prescribed nonconstant mean curvature in Riemannian manifolds

Given a function f on a smooth Riemannian manifold without boundary, we prove that if p∈M is a non-degenerate critical point of f, then a neighborhood of p contains a foliation by spheres with mean curvature proportional to f. This foliation is essentially unique. The nondegeneracy assumption can be substantially relaxed, at the expense of losing the property that the family of spheres with prescribed mean curvature defines a foliation.

Non-constant geometric curvature for tailored spin–orbit coupling and chirality in superconductor-magnet heterostructures

We show that tailoring the geometric curvature profile of magnets can be used for bespoke design of an effective non-relativistic spin–orbit coupling, which may be used to control proximity effects if the magnet is coupled to a superconductor. We consider proximity-coupled one-dimensional magnetic wires with variable curvatures, specifically three distinct shapes classified as J-, C-, and S-type. We demonstrate a chirality-dependent spin polarization of the superconducting correlations, and show the role of curvature in determining the ground state of mixed-chirality junctions. We speculate on how this may be implemented in novel device design, and include analysis of its usage in a spin-triplet SQUID.

Characterization of Out-of-Plane Curved Fluidic Oscillators

A novel fluidic oscillator design, in which the oscillator is curved along the primary flow direction, was evaluated. The effects of mass flow rate, hydraulic diameter, aspect ratio, curvature radius, surface roughness, inlet orientation, and nonconstant curvatures were experimentally investigated. Measurements of the oscillation Strouhal number, discharge coefficient, spreading angle, and sweeping angle were used to characterize the jet. The resulting oscillators were found to fall into one of three categories: oscillated at the same frequency and sweep angle as conventional flat oscillators; oscillated at a slightly higher frequency and lower sweep angle than flat oscillators; or no dominant oscillation frequency detected and with no sweeping action. An unsteady Reynolds-averaged Navier–Stokes computational fluid dynamics simulation revealed fundamental differences in the internal flow mechanisms between flat and curved oscillators that drive the sweeping jet. The curvature between 37.5 and 62.5% of the total length, or the region from the inlet nozzle to halfway through the main chamber, was a primary factor influencing the response type of a design. Due to the curvature of these oscillators, they have the ability to be used in geometrically constrained spaces, such as the leading edge of wings and turbine vanes.

Early-stage modeling and analysis of continuum compliant structure for multi-DOF endoluminal forceps using pseudo-rigid-body model

Early detection and treatment of intraluminal diseases enable minimally invasive surgery and can lead to a high cure rate. Advanced devices with multiple degrees of freedom (DOFs) make narrow intraluminal procedures easier and safer. In a previous study, we developed a multi-DOF compliant endoluminal forceps with a tendon-sheath mechanism. The maximum bending stress of this design was reduced by changing the thickness of a compliant hinge in each segment, and the forceps achieved a wide range of motion. However, its deformed shapes with a non-constant curvature, due to a compliant inhomogeneous-thickness hinge structure, hinder fine manipulation in a narrow lumen. Here, we construct an accurate model of this compliant inhomogeneous-thickness hinge structure using pseudo-rigid-body model. In the proposed method, each compliant hinge is represented by a torsion spring and rotational joint, and the deformed shape is estimated from the bending angle caused by the tensile force. We derive the coefficient of dynamic friction by considering the friction between the wire and compliant joint based on a belt friction model. These novel calculations allow to consider individual differences in material properties and surface roughness. We experimentally confirm the feasibility of constructing a highly accurate model with a lower calculation cost than the finite element method. Our proposal seems suitable for developing dexterous forceps and other endoluminal devices, such as catheters, which mitigate operation errors and help approach lesions in narrow lumens.

Efficient cutting path planning for a non-spherical tool based on an iso-scallop height distance field

Iso-scallop height machining means, when machining a freeform surface, the scallop height between any two neighboring tool paths on the surface will be a constant (i.e., the given threshold), which is preferable among various freeform surface machining strategies due to its high machining efficiency as well as better machine tool’s dynamics. However, all the existing iso-scallop height path planning methods pertain to only the ball-end or flat-end types of tools. In recent years, the non-spherical cutting tool has become more and more popular, especially for five-axis machining of complex freeform surfaces, majorly owing to its non-constant curvature which can be utilized to adaptively fit the tool to the surface to both avoid the local gouging and enlarge the cutting width. However, there have been no reported works on iso-scallop height five-axis tool path generation for a non-spherical tool, and, in this paper, we present one. Specifically, we first define and construct two fields on the surface to be machined-the collision-free tool orientation field (vector) and the iso-scallop height distance field (scalar). The iso-lines of the scalar field and their associated tool orientation field vectors then naturally serve as potential iso-scallop height five-axis tool paths, and we present a propagation-based algorithm to construct the desired tool path from the iso-lines. The computer simulation and physical cutting experiments confirm that everywhere on the surface, except maybe near the saddle curves of the scalar filed, the scallop height is exactly the given threshold. By adding the saddle curves as extra tool paths, the final machined surface then is assured of the required scallop height requirement.

The Dunkl oscillator on a space of nonconstant curvature: An exactly solvable quantum model with reflections

We introduce the Dunkl-Darboux III oscillator Hamiltonian in N dimensions as a λ−deformation of the Dunkl oscillator. This deformation is interpreted as the introduction of a non-constant curvature on the underlying space or, equivalently, as a quadratic position-dependent mass for the Dunkl oscillator. This new ND quantum model is shown to be exactly solvable, and its eigenvalues and eigenfunctions are explicitly presented. It is shown that in the 2D case both Darboux III and Dunkl oscillators can be coupled with a constant magnetic field, thus giving rise to two new exactly solvable quantum systems in which the effect of the λ−deformation and of the Dunkl derivatives on the Landau levels can be studied. Finally, the full 2D Dunkl-Darboux III oscillator is coupled with the magnetic field and shown to define an exactly solvable Hamiltonian, where the interplay between the λ−deformation and the magnetic field is explicitly illustrated.

Exploring accelerating hairy black holes in 2+1 dimensions: the asymptotically locally anti-de Sitter class and its holography

In the realm of lower-dimensional accelerating spacetimes, it is well-established that the presence of domain walls, which are co-dimension one topological defects, is a necessary condition for their construction. We expand upon the geometric framework employed in the generation of such spacetime solutions by incorporating a conformally coupled scalar field within the matter sector. This endeavor leads to the identification of several new families of three-dimensional accelerating spacetimes with asymptotically locally anti-de Sitter (AdS) behavior. Notably, one of these solutions showcases a hairy generalization of the accelerating BTZ black hole. This solution is constructed at both slow and rapid phases of acceleration, and its connection with established vacuum spacetime models is explicitly elucidated. The inclusion of the scalar field imparts a non-constant Ricci curvature to the domain wall, thereby rendering these configurations particularly suitable for the construction of two-dimensional quantum black holes. To establish a well-posed variational principle in the presence of the domain wall, two essential steps are undertaken. First, we extend the conventional renormalized AdS3 action to accommodate the presence of the scalar field. Second, we explicitly incorporate the Gibbons-Hawking-York term associated with the internal boundaries of our geometries and account for the tension of the domain wall in the action. This dual step process enables us to derive the domain wall field equations via the variational principle. Consequently, the action furnishes the appropriate quantum statistical relation. We engage in holographic computations, thereby determining the explicit form of the holographic stress tensor. In this context, the stress tensor can be expressed as that of a perfect fluid situated on a curved background. Additionally, it paves the road to ascertain the spacetime mass. Finally, we close by demonstrating the existence of three-dimensional accelerating spacetimes with asymptotically locally flat and asymptotically locally de Sitter geometries, particularly those embodying black holes.

An Improved Static Model for Bidirectional Notched Continuum Robot Considering the Cable Tension Loss

Abstract The notched continuum mechanism is particularly suitable for natural orifice transluminal surgery benefiting from its small size and hollow structure. However, the widely used kinematic model based on constant curvature assumption does not reveal the actual deformation of the continuum mechanism, and its control accuracy is unstable, while the general mechanics model has the problem that the tension of the distal driving cable is difficult to measure. In this paper, a nonconstant curvature static model for a bidirectional V-shaped notched continuum mechanism is presented. The deformation of each part of the continuum mechanism from the distal end to the proximal end is analyzed in turn. The tension loss of the driving cable caused by the contact with the continuum mechanism is modeled using the capstan equation. The recursive equation between the deformation of each part of the continuum mechanism from the proximal end is derived, which can be solved numerically. The bending state of the continuum mechanism can then be estimated when only the tension of the proximal flexible cable is known. The model is experimentally verified by driving the continuum mechanism to move at a very low speed. The experiment results show that the estimation effect of the proposed model is significantly improved compared with that of the constant curvature model.

Extended-depth-of-focus wavefront design frompseudo-umbilical space curves.

Designing extended-depth-of-focus wavefronts is required in multiple optical applications. Caustic location and structure analysis offer a powerful tool for designing such wavefronts. An intrinsic limitation of designing extended-depth-of-focus wavefronts is that any smooth surface, with a non-constant mean curvature, unavoidably introduces a separation between caustic sheets, which is proportional to the ratio of change of the mean curvature along a curve embedded in the wavefront. We present a method to obtain extended-depth-of-focus wavefronts where the mean curvature variation ratio is reduced thanks to using a long circle-involute space curve effectively filling the wavefront surface. Additionally, we present a variant of the method in which the wavefront is modified within a small tubular neighborhood of the circle involute in order to partially meet the umbilical condition along that tubular region. Finally, we provide some numerical results showing the potential of our method in an application example.

Translation-Like Isoptic Surfaces and Angle Sums of Translation Triangles in textbf{Nil} Geometry

After having investigated the geodesic and translation triangles and their angle sums in textbf{Sol} and widetilde{{textbf{S}}{textbf{L}}_2{textbf{R}}} geometries we consider the analogous problem in textbf{Nil} space that is one of the eight 3-dimensional Thurston geometries. We analyze the interior angle sums of translation triangles in textbf{Nil} geometry and we provide a new approach to prove that it can be larger than or equal to pi . Moreover, for the first time in non-constant curvature Thurston geometries we have developed a procedure for determining the equations of textbf{Nil} isoptic surfaces of translation-like segments and as a special case of this we examine the textbf{Nil} translation-like Thales sphere, which we call Thaloid. In our work we will use the projective model of textbf{Nil} described by Molnár (Beitr Algebra Geom 38(2):261–288, 1997).

Three-dimensional nonlinear dynamics of a thin liquid film on a spinning ellipsoid

The present work investigates the three-dimensional flow of a thin liquid film distributed on the outer surface of an ellipsoid, rotating around the vertical axis at constant angular velocity. The lubrication approximation expressing the evolution of the film thickness, originally developed for stationary curved substrates, has been re-derived by including the non-inertial forces associated with the rotation. This comprehensive model, which incorporates the gravitational, centrifugal, and capillary forces, is employed for a parametric investigation via numerical simulations. The results validate and extend the conclusions of our former study covering the axisymmetric case and bring about an advanced understanding by exploring non-axisymmetric effects. The parametric analysis sheds light on the significance of rotation on a non-constant curvature substrate by comparing the thickness profiles with the static case.

Exotic black hole thermodynamics in third-order Lovelock gravity

The generalisation of Birkhoff’s theorem to higher dimensions in Lovelock gravity allows for black hole solutions with horizon geometries of non-constant curvature. We investigate thermodynamic aspects of these ‘exotic’ black hole solutions, with a particular emphasis on their phase transitions. We consider an extended phase space where the cosmological constant acts as a thermodynamic pressure, and examine both uncharged and U(1) charged solutions. In d = 7, black hole solutions are restricted to having constant curvature horizon base manifolds. Uncharged d = 7 black holes possess novel triple point phenomena analogous to those recently uncovered in exotic d = 6 black holes in Gauss–Bonnet gravity, while their charged counterparts generically undergo small-large black hole phase transitions. In d = 8, we find that both charged and uncharged black holes exhibit triple point behaviour and small-large black hole transitions. We also show that a wide range of ‘exotic’ horizon geometries can be ruled out due to the appearance of naked singularities.

Design and Fabrication of a Fiber Bragg Grating Shape Sensor for Shape Reconstruction of a Continuum Manipulator.

Continuum dexterous manipulators (CDMs) are suitable for performing tasks in a constrained environment due to their high dexterity and maneuverability. Despite the inherent advantages of CDMs in minimally invasive surgery, real-time control of CDMs' shape during nonconstant curvature bending is still challenging. This study presents a novel approach for the design and fabrication of a large deflection fiber Bragg grating (FBG) shape sensor embedded within the lumens inside the walls of a CDM with a large instrument channel. The shape sensor consisted of two fibers, each with three FBG nodes. A shape-sensing model was introduced to reconstruct the centerline of the CDM based on FBG wavelengths. Different experiments, including shape sensor tests and CDM shape reconstruction tests, were conducted to assess the overall accuracy of the shape-sensing. The FBG sensor evaluation results revealed the linear curvature-wavelength relationship with the large curvature detection of 0.045 mm and a high wavelength shift of up to 5.50 nm at a 90° bending angle in both the bending directions. The CDM's shape reconstruction experiments in a free environment demonstrated the shape-tracking accuracy of 0.216 ± 0.126 mm for positive/negative deflections. Also, the CDM shape reconstruction error for three cases of bending with obstacles was observed to be 0.436 ± 0.370 mm for the proximal case, 0.485 ± 0.418 mm for the middle case, and 0.312 ± 0.261 mm for the distal case. This study indicates the adequate performance of the FBG sensor and the effectiveness of the model for tracking the shape of the large-deflection CDM with nonconstant-curvature bending for minimally invasive orthopedic applications.

Beyond Schwarzschild–de Sitter spacetimes. III. A perturbative vacuum with nonconstant scalar curvature in R+R2 gravity

In violation of the generalized Lichnerowicz theorem advocated by Nelson and others, quadratic gravity admits vacua with non-constant scalar curvature. In a recent publication [Phys. Rev. D 106, 104004 (2022)], we revitalized a program that Buchdahl originated but prematurely abandoned circa 1962 [Nuovo Cimento 23, 141 (1962)], and uncovered a novel exhaustive class of static spherically symmetric vacua for pure $R^2$ gravity. The Buchdahl-inspired metrics we obtained therein are exact solutions which exhibit non-constant scalar curvature. A product of fourth-order gravity, the metrics entail a new (Buchdahl) parameter $k$ which allows the Ricci scalar to vary on the manifold. The metrics are able to defeat the generalized Lichnerowicz theorem by evading an overly strong restriction on the asymptotic falloff in the Ricci scalar as assumed in the theorem. The Buchdahl parameter $k$ is a new characteristic of pure $R^2$ gravity, a higher-derivative theory. By venturing that the Buchdahl parameter should be a universal hallmark of higher-derivative gravity at large, in this paper we seek to extend the concept to the quadratic action $R^2+\gamma\,(R-2\Lambda)$. We are able to determine that the quadratic field equation admits a perturbative vacuo that is valid up to the order $O(k^2)$. Conforming with our guiding intuition, the Ricci scalar is non-constant, including the asymptotically flat case, as long as $k\neq0$ and $\gamma\neq0$. The existence of such an asymptotically flat vacuo with non-constant scalar curvature defeats the generalized Lichnerowicz theorem in its entirety. Our finding thus warrants restoring the $R^2$ term in the full quadratic action, $\gamma\,R+\beta\,R^2-\alpha\,C^{\mu\nu\rho\sigma}C_{\mu\nu\rho\sigma}$, when applying the L\"u-Perkins-Pope-Stelle ansatz. Implications to the L\"u-Perkins-Pope-Stelle solution are discussed herein.