Curvature Form Research Articles

Non-Parametric Shape Design of Free-Form Shells Using Fairness Measures and Discrete Differential Geometry

A non-parametric approach is proposed for shape design of free-form shells discretized into triangular mesh. The discretized forms of curvatures are used for computing the fairness measures of the surface. The measures are defined as the area of the offset surface and the generalized form of the Gauss map. Gaussian curvature and mean curvature are computed using the angle defect and the cotangent formula, respectively, defined in the field of discrete differential geometry. Optimization problems are formulated for minimizing various fairness measures for shells with specified boundary conditions. A piecewise developable surface can be obtained without a priori assignment of the internal boundary. Effectiveness of the proposed method for generating various surface shapes is demonstrated in the numerical examples.

Kähler immersions of Kähler–Ricci solitons into definite or indefinite complex space forms

Let $(g, X)$ be a K\"ahler-Ricci soliton on a complex manifold $M$. We prove that if the K\"ahler manifold $(M, g)$ can be K\"ahler immersed into a definite or indefinite complex space form of constant holomorphic sectional curvature $2c$, then $g$ is Einstein. Moreover, its Einstein constant is a rational multiple of $c$.

Kerr–de Sitter black hole revisited

We interpret the cosmological constant as the energy of the vacuum, and under a minimum amount of assumptions, we show that it is deformed in the vicinity of a black hole. This leads us to reexamine the Kerr-de Sitter solution. We provide a new solution, simpler and geometrically richer, which shows the impact of the rotation in form of a warped curvature. We carry out a detailed and exact study on the new black hole solution, and we conclude with a conjecture regarding the possible impact of our results on alternative theories.

Real hypersurfaces of complex space forms satisfying Fischer–Marsden equation

Let M be a real hypersurface of a complex space form of constant curvature c. In this paper, we study the hypersurface M which admits a nontrivial solution to Fischer–Marsden equation, that is, the induce metric g of M satisfies $$ Hess _g(\nu )=(\varDelta _g \nu )g+\nu S_g$$ , where $$\nu $$ is a nontrivial function. We prove that there does not exist a complete Hopf real hypersurface in a non-flat complex space form satisfying Fischer–Marsden equation. Finally, we show that a complete real hypersurface with $$A\xi =\alpha \xi $$ , $$\alpha \ne 0$$ , of a complex Euclidean space $${\mathbb {C}}^n$$ satisfying Fischer–Marsden equation is locally congruent to a sphere or $${\mathbb {S}}^1 \times {\mathbb {R}}^{2n-2}$$ .

Gravitational waves on Earth and their warming effects

The gravity or reactive bundle energy is the outlet of the morphogenetic impact, known as “BIG BANG”, creating a bounded ordered/structured universe along with the solar system, including the EARTH-world with its human race. Post-impact, the huge kinetic energy is spread into stellar bodies associated with the light flux under strong mutual connections or gravitational bundle. Einstein’s general relativity theory including the gravitational field can be expressed under a condensed tensor formulation as E  R − Rg =  T where E defines the geometry via a curved space-time structure (R) over the gravity field (1/2Rg), embedded in a matter distribution T The fundamental (ten non-linear partial differential) equations of the gravitational field are a kind of the space-time machine using the curvature of a four-dimensional space-time to engender the gravity field carrying away material structures. Gravity according to the curved space-time theory is not seen as a gravitational force, but it manifests itself in the relativistic form of the space-time curvature needing the constancy of the light speed. But the constant light velocity makes the tidal wave/pulsating energy, a characteristic of solar energy, impossible. The Einstein’s field equation, expressed in terms of tensor formulation along with the constant light speed postulate, needs two special space-time tensors (curvature and torsion) in 4 dimensions, where for the simplicity the torsion/twist tensor is less well approximated (Bianchi identity) leading to a constant/frozen gravity (twist-free gravity).The non-zero torsion tensor plays a significant physical role in the planetary dynamics as a finest gear of a planet, where its spinning rotation is directly connected to the own work and space-time structure (or clock), controlled by light fluctuations (or tidal effect of gravity). The spin correction of Einstein’s gravitational field refers to the curvature-torsion effect coupled with fluctuating light speed. The mutual curvature-torsion bundle self-sustained by the quantum fluctuations of light speed engenders helical gravitational wave fields of a quantum nature where bodies orbit freely in the light speed field (cosmic wind). In contrast to the Einstein’s field equation describing a gravitational frozen field, a quantum tidal gravity model is proposed in the paper.

CR Maximal Dimensional Submanifolds of Kenmotsu Space Forms

For n > 5, we investigate (n + 1)-dimensional contact CR-submanifolds M of (n − 1) contact CR-dimension in a complete simply connected Kenmotsu space form of constant ϕ-holomorphic sectional curvature c which satisfy the condition h(FX,Y ) + h(X,FY ) = 0 for any vector fields X,Y tangent to M, where h and F denote the second fundamental form and a skew-symmetric endomorphism acting on the tangent space of M, respectively.

Higher-dimensional Euler fluids and Hasimoto transform: counterexamples and generalizations

The binormal (or vortex filament) equation provides the localized induction approximation of the 3D incompressible Euler equation. We present explicit solutions of the binormal equation in higher-dimensions that collapse in finite time. The local nature of this phenomenon suggests a possibility of the singularity appearance in nearby vortex blob solutions of the Euler equation in 5D and higher. Furthermore, the Hasimoto transform takes the binormal equation to the NLS and barotropic fluid equations. We show that in higher dimensions the existence of such a transform would imply the conservation of the Willmore energy in skew-mean-curvature flows and present counterexamples for vortex membranes based on products of spheres. These (counter)examples imply that there is no straightforward generalization to higher dimensions of the 1D Hasimoto transform. We derive its replacement, the evolution equations for the mean curvature and torsion form for membranes, thus generalizing the barotropic fluid and Da Rios equations.

On the Canonical Foliation of an Indefinite Locally Conformal Kähler Manifold with a Parallel Lee Form

We study the semi-Riemannian geometry of the foliation F of an indefinite locally conformal Kähler (l.c.K.) manifold M, given by the Pfaffian equation ω=0, provided that ∇ω=0 and c=∥ω∥≠0 (ω is the Lee form of M). If M is conformally flat then every leaf of F is shown to be a totally geodesic semi-Riemannian hypersurface in M, and a semi-Riemannian space form of sectional curvature c/4, carrying an indefinite c-Sasakian structure. As a corollary of the result together with a semi-Riemannian version of the de Rham decomposition theorem any geodesically complete, conformally flat, indefinite Vaisman manifold of index 2s, 0<s<n, is locally biholomorphically homothetic to an indefinite complex Hopf manifold CHsn(λ), 0<λ<1, equipped with the indefinite Boothby metric gs,n.

Reconstructing wormhole solutions in curvature based Extended Theories of Gravity

Static and spherically symmetric wormhole solutions can be reconstructed in the framework of curvature based Extended Theories of Gravity. In particular, extensions of the General Relativity, in metric and curvature formalism give rise to modified gravitational potentials, constituted by the classical Newtonian potential and Yukawa-like corrections, whose parameters can be, in turn, gauged by the observations. Such an approach allows to reconstruct the spacetime out of the wormhole throat considering the asymptotic flatness as a physical property for the related gravitational field. Such an argument can be applied for a large class of curvature theories characterising the wormholes through the parameters of the potentials. According to this procedure, possible wormhole solutions could be observationally constrained. On the other hand, stable and traversable wormholes could be a direct probe for this class of Extended Theories of Gravity.

Embeddedness, convexity, and rigidity of hypersurfaces in product spaces

We establish the following Hadamard–Stoker-type theorem: Let \(f:M^n\rightarrow \mathscr{H} ^{\,\,\, n}\times \mathbb{R}\) be a complete connected hypersurface with positive definite second fundamental form, where \(\mathscr{H} ^{\,\,\, n}\) is a Hadamard manifold. If the height function of f has a critical point, then it is an embedding and M is homeomorphic to \(\mathbb{S}^n\) or \(\mathbb{R}^n.\) Furthermore, f(M) bounds a convex set in \(\mathscr{H} ^{\,\,\, n}\times \mathbb{R}.\) In addition, it is shown that, except for the assumption on convexity, this result is valid for hypersurfaces in \(\mathbb{S}^n\times \mathbb{R}\) as well. We apply these theorems to show that a compact connected hypersurface in \(\mathbb{Q}_{\epsilon}^{n}\times \mathbb{R}\) (\(\epsilon =\pm 1\)) is a rotational sphere, provided it has either constant mean curvature and positive-definite second fundamental form or constant sectional curvature greater than \((\epsilon +1)/2.\) We also prove that, for \(\bar{M}=\mathscr{H} ^{\,\,\, n} \,{\mathrm{or}} \,\, \mathbb{S}^n,\) any connected proper hypersurface \(f:M^n\rightarrow \bar{M}^n \times \mathbb{R}\) with positive semi-definite second fundamental form and height function with no critical points is embedded and isometric to \(\Sigma ^{n-1}\times \mathbb{R},\) where \(\Sigma ^{n-1}\subset \bar{M}^n\) is convex and homeomorphic to \(\mathbb{S}^{n-1}\) (for \(\bar{M}^n=\mathscr{H} ^{\,\,\, n}\) we assume further that f is cylindrically bounded). Analogous theorems for hypersurfaces in warped product spaces \(\mathbb{R}\times _\varrho \mathscr{H} ^{n}\) and \(\mathbb{R}\times _\varrho \mathbb{S}^{n}\) are obtained. In all of these results, the manifold \(M^{n}\) is assumed to have dimension \(n\ge 3.\)

Geometric estimates on weighted p-fundamental tone and applications to the first eigenvalue of submanifolds with bounded mean curvature

ABSTRACT This paper generalizes to the context of smooth metric measure spaces and submanifolds with negative sectional curvatures some well-known geometric estimates on the p-fundamental tone by using vector fields satisfying a positive divergence condition. Choosing the vector field to be the gradient of an appropriately chosen distance function yields generalised McKean estimates whilst other choices of vector fields yield new geometric estimates generalising certain results of Lima et al. (Nonlinear Anal. 2010;72:771–781). We also obtain a lower bound on the spectrum of the weighted p-Laplacian on a complete noncompact smooth metric space with the underlying space being a submanifold with bounded mean curvature in the hyperbolic space form of constant negative sectional curvature generalising results of Du and Mao (J Math Anal Appl. 2017;456:787–795).

On constant curvature submanifolds of space forms

We prove a converse to well-known results by E. Cartan and J. D. Moore. Let f:Mcn→Qc˜n+p be an isometric immersion of a Riemannian manifold with constant sectional curvature c into a space form of curvature c˜, and free of weak-umbilic points if c>c˜. We show that the substantial codimension of f is p=n−1 if, as shown by Cartan and Moore, the first normal bundle possesses the lowest possible rank n−1. These submanifolds are of a class that has been extensively studied due to their many properties. For instance, they are holonomic and admit Bäcklund and Ribaucour transformations.

НЕПОСРЕДСТВЕННЫЕ РЕЗУЛЬТАТЫ УРЕТРОПЛАСТИКИ ПРИ РАЗНЫХ ФОРМАХ ГИПОСПАДИИ

Objective: To summarize a ten-year experience of urethroplasty in various forms of hypospadias. Methods: The results of surgical treatment of 165 patients with hypospadias aged 2-25 years operated at the Maiwand Hospital in Kabul were analyzed. More than half of the patients (n=95; 57.6%) were residents of Kabul province, while the remaining patients (n=60; 42.4%) came from other provinces of Afghanistan. The following forms of hypospadias were identified: glanular (n=40; 24.2%), coronal (n=45; 27.3%), mid-penile (n=51; 30.9%), and scrotal (n=29; 17, 6%). Results: Most of the patients were operated using the Salinas method (n=30; 18.2%), Dupley repair (n=40; 24.2%), and their combinations (n=20; 12.1%) methods. Patients with severe penile curvature, scrotal and perineal forms of hypospadias underwent the two-stage operations. General anesthesia was mainly used for surgery, while local anesthesia with neuroleptanalgesia was applied in patients aged 15 years and above. In all cases, optical magnification and microsurgical technique were used. Satisfactory results were obtained in 140 (84.8%) patients. Urethro-cutaneous fistulas developed in 20 (12.2%) patients. Marginal flap necrosis occurred in 2 (1.2%) cases, and partial wound suppuration occurred in 3 (1.8%) cases. There were no lethal outcomes.Conclusion: The use of optical magnification and precision technique made it possible to achieve satisfactory results in the early postoperative period in 84.8% of patients. Our observations showed that circumcision in children should only be performed by an experienced surgeon, since in most cases a novice surgeon can miss the presence of hypospadias. Patients diagnosed with hypospadias, should be referred to specialized centers where successful urethroplasty operations may be performed. Keywords: Hypospadias, glanular form, coronal form, mid-penile form, scrotal form, urethroplasty, urethro-cutaneous fistula.

Damage analysis and assessment of the impact of damage on the operation of supporting structures of oil refining tube furnaces

Relevance. Due to the widespread use in practice, tubular furnaces were chosen as the object of study of this work. The article provides an analysis of damage to the supporting structures of oil refining tubular furnaces. The causes of damage and the physical nature of the development of damage are established. According to the results of field surveys, it was found that about 10% of the furnaces are operated with damage in the form of significant curvature of the supporting structures that developed as a result of the explosion of the gas-air mixture and the technological product inside the furnace space. The aim of the work is to analyze the damage and assess the impact of damage on the operation of the supporting structures of furnaces. Methods. The main research results were obtained by static numerical analysis of spatial models of furnace frameworks in the LIRA-SAPR software package. This complex belongs to the class of software products that implement the finite element method. Results. According to the results of calculating a series of models of structures, the effect of damage on the operation of the supporting structures of the furnace is determined. Based on the analysis of calculation data for models of tube furnaces with damage and comparison of calculation results for furnace models with structural damage identified during the survey, ways to optimize the design decisions of an industrial furnace are determined. Studies have shown the need to improve the design of tube furnaces in the direction of improving technology and improving the structural form of the supporting frame of the furnace.

A fully coupled ALE interface tracking method for a pressure-based finite volume solver

Simulation of two-phase gas-liquid flows is a challenging problem in terms of predicting the interface position and appropriately coupling the phases. Stability restrictions induced by surface tension of the liquid phase may increase the level of difficulty of the simulation. The Arbitrary Lagrangian-Eulerian (ALE) method along with an interface tracking technique is an approach for a precise prediction of the interface position. The restrictions in the simulation due to surface tension necessitate an implicitly coupled solution algorithm. In this research, the interface kinematic condition equation is discretized in a new coupled form, including an interface displacement variable as well as the interface velocity components. Furthermore, an implicitly discretized formulation of interface curvature is implemented in the interface normal force balance to facilitate the complete coupling of the interface displacement movement to the hydrodynamic behaviour of the flow. Finally, the governing equations of both phases as well as complete set of interface equations are solved simultaneously in a system of linearized algebraic equations. A partially coupled interface tracking (PCIT) method and a fully coupled interface tracking method (FCIT) are developed and evaluated in predictions of a backward-facing step flow, of liquid falling films with and without interaction with a gas phase flow, of an oscillating drop, and of a rising bubble. The results show that in the viscocapillary regime, the FCIT method keeps its stability in a wide range of CFL numbers, whereas the PCIT method is stable only for CFL ≤1. When the surface tension is ignored in a backward-facing step flow, the PCIT method also remains stable for higher CFL number due to the coupled formulation of interface displacement and slope. The present results are in excellent agreement with previous numerical and experimental work results reported in the literature.

On the Principal Curvatures of Complete Minimal Hypersurfaces in Space Forms

In recent decades, there has been an increase in the number of publications related to the hypersurfaces of real space forms with two principal curvatures. The works focus mainly on the case when one of the two principal curvatures is simple. The purpose of this paper is to study a slightly more general class of complete minimal hypersurfaces in real space forms of constant curvature c, namely those with $$\mathrm{n}-1$$ principal curvatures having the same sign everywhere. From assumptions on the scalar curvature R and the Gauss–Kronecker curvature K we characterize Clifford tori if $$c > 0$$ and prove that K is identically zero if $$c \le 0$$ .

Minimal submanifolds from the abelian Higgs model

Given a Hermitian line bundle Lrightarrow M over a closed, oriented Riemannian manifold M, we study the asymptotic behavior, as epsilon rightarrow 0, of couples (u_epsilon ,nabla _epsilon ) critical for the rescalings Eϵ(u,∇)=∫M(|∇u|2+ϵ2|F∇|2+14ϵ2(1-|u|2)2)\\documentclass[12pt]{minimal}\t\t\t\t\\usepackage{amsmath}\t\t\t\t\\usepackage{wasysym}\t\t\t\t\\usepackage{amsfonts}\t\t\t\t\\usepackage{amssymb}\t\t\t\t\\usepackage{amsbsy}\t\t\t\t\\usepackage{mathrsfs}\t\t\t\t\\usepackage{upgreek}\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\t\t\t\t\\begin{document}$$\\begin{aligned} E_\\epsilon (u,\\nabla )=\\int _M\\Big (|\\nabla u|^2+\\epsilon ^2|F_\\nabla |^2+\\frac{1}{4\\epsilon ^2}(1-|u|^2)^2\\Big ) \\end{aligned}$$\\end{document}of the self-dual Yang–Mills–Higgs energy, where u is a section of L and nabla is a Hermitian connection on L with curvature F_{nabla }. Under the natural assumption limsup _{epsilon rightarrow 0}E_epsilon (u_epsilon ,nabla _epsilon )<infty , we show that the energy measures converge subsequentially to (the weight measure mu of) a stationary integral (n-2)-varifold. Also, we show that the (n-2)-currents dual to the curvature forms converge subsequentially to 2pi Gamma , for an integral (n-2)-cycle Gamma with |Gamma |le mu . Finally, we provide a variational construction of nontrivial critical points (u_epsilon ,nabla _epsilon ) on arbitrary line bundles, satisfying a uniform energy bound. As a byproduct, we obtain a PDE proof, in codimension two, of Almgren’s existence result for (nontrivial) stationary integral (n-2)-varifolds in an arbitrary closed Riemannian manifold.

Compositional and Artistic Territory Assessment of the Kyiv Cultural Landscape Monument

The purpose of the study is to carry out a compositional and artistic territory’s evaluation of the Landscape Monument “Historic Landscape of the Kyiv Mountains and the Dnieper River Valley”, with recommendations on how to preserve the traditional character of the environment as well as the environment itself. The research methodology is based on the use of general scientific and special methods, in particular applied methods of theoretical research, namely the historical method in a systematic approach, when the object of research is considered in its origin and development. The systematic approach is to comprehensively study the object as a certain unity with the coherent functioning of the parts and the whole. Methods of bibliographic, archival, cartographic, iconographic research, as well as methods of empirical research: observations, full-scale studies, comparisons and mapping, have been applied. The scientific novelty of the research is that for the first time a compositional and artistic territory evaluation of the landmark “Historic Landscape of the Kyiv Mountains and the Dnieper River Valley” was carried out. This expertise should form the basis for defining / adjusting the boundaries and modes of use of the site itself and its conservation areas. The findings and conclusions of the study are based on the expert evaluation of the territory of the landmark “Historic landscape of the Kyiv Mountains and the Dnieper River valley” in the compositional and artistic aspect. It is concluded that the entire territory is of high value, because it is a landmark that is currently still a part of to the local accounting category, but by its importance for Ukrainian history and culture it should be entered in the State Register of Immovable Monuments of Ukraine as a monument of national importance. However, despite such a high overall assessment of the entire territory of the monument, two categories of territories can be distinguished within it. The most valuable of them are on the high marks of the Dnieper plateau, with ensembles and complexes of town-planning, architectural, historical, archeological monuments of national importance and UNESCO World Heritage Site – the Kyiv- Pechersk Lavra ensemble. The relatively less valuable undeveloped territories in the Dnieper valley serve as a visual pool with many viewing points and fronts and a kind of pedestal for large-scale forms of positive curvature topped by architectural and urban dominants.

A note on equivariant biharmonic maps and stable biharmonic maps

In this note, we generalize biharmonic equation for rotationally symmetric maps ([4], [20], [12]) to equivariant maps between model spaces and use it to give a complete classification of rotationally symmetric conformal biharmonic maps from a 4-dimensional space form into a 4-dimensional model space. We also give an improved second variation formula for biharmonic maps into a space form and use it to prove that there exists no stable proper biharmonic map with constant square norm of tension field from a compact Riemannian manifold without boundary into a space form of positive sectional curvature.

PSEUDO-PARALLEL KAEHLERIAN SUBMANIFOLDS IN COMPLEX SPACE FORMS

Let $\tilde{M}^{m}(c)$ be a complex $m$-dimensional space form of holomorphic sectional curvature $c$ and $M^{n}$ be a complex $n$-dimensional Kaehlerian submanifold of $\tilde{M}^{m}(c).$ We prove that if $M^{n}$ is pseudo-parallel and $Ln-\frac{1}{2}(n+2)c\geqslant 0$ then $M$ $^{n}$ is totally geodesic. Also, we study Kaehlerian submanifolds of complex space form with recurrent second fundamental form.