Curvature Form Research Articles

On a class of immersions of spheres into space forms of nonpositive curvature

Let \( M^{n+1} \) (\( n \ge 2 \)) be a simply-connected space form of sectional curvature \( -\kappa ^2 \) for some \( \kappa \ge 0 \), and I an interval not containing \( [-\kappa ,\kappa ] \) in its interior. It is known that the domain of a closed immersed hypersurface of M whose principal curvatures lie in I must be diffeomorphic to the n-sphere \( \mathbb {S}^n \). These hypersurfaces are thus topologically rigid. The purpose of this paper is to show that they are also homotopically rigid. More precisely, for fixed I, the space \( \mathcal {F} \) of all such hypersurfaces is either empty or weakly homotopy equivalent to the group of orientation-preserving diffeomorphisms of \( \mathbb {S}^n \). An equivalence assigns to each element of \( \mathcal {F} \) a suitable modification of its Gauss map. For M not simply-connected, \( \mathcal {F} \) is the quotient of the corresponding space of hypersurfaces of the universal cover of M by a natural free proper action of the fundamental group of M.

Sizing High-Aspect-Ratio Wings with a Geometrically Nonlinear Beam Model

This study considers the effect of geometric nonlinearity at the design concept stage of an aircraft wing. A nonlinear finite element beam model adopting the finite volumes concept with an intrinsic strain and curvature formulation is used to size a single-aisle passenger aircraft. A linearized version of this geometrically nonlinear formulation provides a linear benchmark from which the nonlinear predictions of loads, weight, and performance can be compared. A baseline study on a wing with an aspect ratio of 18 shows that geometric nonlinearity can have a significant impact on the internal loads, leading to a reduction in the wing weight. The effect of aspect ratio is also explored, yielding optimal values at which the Breguet range is maximized and showing the effect of geometric nonlinearity on that range. This is complemented by a parameter study on the effect of varying the wing span and surface area, showing that, if gate limitations in the form of span constraints are imposed on the sizing, the optimal aspect ratio is significantly reduced and geometrically nonlinear effects are mitigated. The paper demonstrates the correlation between geometric nonlinearity and the improvements in the wing mass and Breguet range as compared to a conventional linear analysis.

Forms and Chern Classes on Hermitian Lie Algebroids

In this paper, we study Hermitian Lie algebroids and introduce the notion of Chern A-connection on Hermitian vector bundles. Then, we prove that on every holomorphic vector bundle there exists a unique Chern A-connection and find its connection form and curvature form. Also, we generalize Weitzenbock’s formula to complex Lie algebroids and prove vanishing theorem for holomorphic sections. Moreover, we extend the Chern classes in complex geometry to Lie algebroids framework.

Symplectic critical surfaces with parallel normalized mean curvature vector in two-dimensional complex space forms

In this paper, we obtain a sufficient and necessary condition for the existence of symplectic critical surfaces with parallel normalized mean curvature vector in two-dimensional complex space forms. Explicitly, we find that there does not exist any symplectic critical surface with parallel normalized mean curvature vector in two-dimensional complex space forms of nonzero constant holomorphic sectional curvature. And there exists and only exists a two-parameters family of symplectic critical surfaces with parallel normalized mean curvature vector in two-dimensional complex plane, which are rotationally symmetric.

Lagrangian Bonnet Problems in Complex Space Forms

In this note we consider Lagrangian Bonnet problem for Lagrangian surfaces in complex space forms. We first give a Bonnet type theorem for conformal Lagrangian surfaces in complex space forms, then we show that any compact Lagrangian surface in the complex space form admits at most one other global isometric Lagrangian surface with the same mean curvature form, unless the Maslov form is conformal. These two Lagrangian surfaces are then called Lagrangian Bonnet pairs. We also studied Lagrangian Bonnet surfaces in complex space forms, and obtain some characterizations of such surfaces.

The geometry of generalized Cheeger–Gromoll metrics on the total space of transitive Euclidean Lie algebroids

Natural metrics (Sasaki metric, Cheeger–Gromoll metric, Kaluza–Klein metrics etc. ) on the tangent bundle of a Riemannian manifold is a central topic in Riemannian geometry. Generalized Cheeger–Gromoll metrics is a family hp,q of natural metrics on the tangent bundle depending on two parameters with p∈R and q≥0. This family possesses interesting geometric properties. If p=q=0 we recover the Sasaki metric and when p=q=1 we recover the classical Cheeger–Gromoll metric. A transitive Euclidean Lie algebroid is a transitive Lie algebroid with a Euclidean product on its total space. In this paper, we show that the analogous of natural metrics 11Natural metrics on the tangent bundles come from a classification of natural transformation of Riemannian metrics on manifolds to metrics on tangent bundles. We use the some terminology in this paper to designate metrics obtained on the total space of a Euclidean Lie algebroid by imitating the expression of g-natural metrics on the tangent bundle.can be built in a natural way on the total space of transitive Euclidean Lie algebroids. Then we study the properties of generalized Cheeger–Gromoll metrics on this new context. We show a rigidity result of these metrics which generalizes some rigidity results known in the case of the tangent bundle. We show also that considering natural metrics on the total space of transitive Euclidean Lie algebroids opens new interesting horizons. For instance, Atiyah Lie algebroids constitute an important class of transitive Lie algebroids and we will show that natural metrics on the total space of Atiyah Euclidean Lie algebroids have interesting properties. In particular, if M is a Riemannian manifold of dimension n, then the Atiyah Lie algebroid associated to the O(n)-principal bundle of orthonormal frames over M possesses a family depending on a parameter k>0 of transitive Euclidean Lie algebroids structures say AO(M,k). When M is a space form of constant curvature c, we show that there exist two constants Cn<0 and K(n,c)>0 such that (AO(M,k),h1,1) is a Riemannian manifold with positive scalar curvature if and only if c>Cn and 0<k≤K(n,c).

Geometric No-Arbitrage Analysis in the Dynamic Financial Market with Transaction Costs

The present paper considers a class of financial market with transaction costs and constructs a geometric no-arbitrage analysis frame. Then, this paper arrives at the fact that this financial market is of no-arbitrage if and only if the curvature 2-form of a specific connection is zero. Furthermore, this paper derives the fact that the no-arbitrage condition for the one-period financial market is equivalent to the geometric no-arbitrage condition. Finally, an example states the equivalence between the geometric no-arbitrage condition and the existence of the solutions for a maximization problem of expected utility.

Interaction of Zinc Oxide and Copper Oxide Nanoparticles with Chlorophyll: A Fluorescence Quenching Study

The present study aims to investigate the interactions of zinc oxide nanoparticles and copper oxide nanoparticles with the major photosynthetic pigment chlorophyll using ultraviolet-visible, steady state, and time resolved laser induced fluorescence spectroscopy. The steady state fluorescence measurements show that zinc oxide and copper oxide nanoparticles quench the fluorescence of chlorophyll in concentration-dependent manner. The Stern-Volmer plot for the chlorophyll-zinc oxide nanoparticles is linear, and the value of quenching constant has been observed to increase with temperature indicating the possibility of dynamic quenching. A decrease in the lifetime of chlorophyll with increase in the concentration of zinc oxide nanoparticles confirms the involvement of dynamic quenching in the chlorophyll–zinc oxide nanoparticle interaction. In the case of copper oxide nanoparticles, the Stern-Volmer plot deviates from linearity observed in the form of upward curvature depicting the presence of both static and dynamic quenching. In addition, the lifetime of chlorophyll decreases with increase in the concentration of copper oxide nanoparticles displaying the dominance of dynamic quenching in the chlorophyll-copper oxide nanoparticle interaction. The decrease observed in the value of binding constant with increasing temperature and negative values of change in enthalpy, entropy, and Gibb’s free energy indicates that van der Waal and hydrogen bonding are the prominent forces during the interaction of chlorophyll with both zinc oxide and copper oxide nanoparticles and that the process is spontaneous and exothermic. The interaction of zinc oxide and copper oxide nanoparticles with chlorophyll occurs through electron transfer mechanism. The obtained results are useful in understanding the sensitization processes involving chlorophyll and zinc oxide and copper oxide nanoparticles.

Kähler geometry of horosymmetric varieties, and application to Mabuchi’s K-energy functional

Abstract We introduce a class of almost homogeneous varieties contained in the class of spherical varieties and containing horospherical varieties as well as complete symmetric varieties. We develop Kähler geometry on these varieties, with applications to canonical metrics in mind, as a generalization of the Guillemin–Abreu–Donaldson geometry of toric varieties. Namely we associate convex functions with Hermitian metrics on line bundles, and express the curvature form in terms of this function, as well as the corresponding Monge–Ampère volume form and scalar curvature. We provide an expression for the Mabuchi functional and derive as an application a combinatorial sufficient condition of properness similar to one obtained by Li, Zhou and Zhu on group compactifications. This finally translates to a sufficient criterion of existence of constant scalar curvature Kähler metrics thanks to the recent work of Chen and Cheng. It yields infinitely many new examples of explicit Kähler classes admitting cscK metrics.

СОВРЕМЕННЫЙ ПОДХОД К ДИАГНОСТИКЕ И ХИРУРГИЧЕСКОМУ ЛЕЧЕНИЮ ИСКРИВЛЕНИЙ ПЕРЕГОРОДКИ НОСА

Objective: Improving the results of septo- and rhinoplasty Methods: The results of surgical treatment of 267 patients with the curvature of the nasal septum without or in combination with the deformation of the external nose are analyzed. The average age of patients was 22.4 years. All variety of forms of curvature of the nasal septum was systematized and conditionally distributed according to the classification of R. Mladina. Isolated curvature of the quadrangular cartilage was found only in 7 cases (2.6%). To determine the type of deformation of the external nose was used Yong Jo Jang’s classification. Septo- and septorhinoplasty in 250 cases (93.6%) was performed by open access. In general, septoplasty combined with rhinoplasty was performed in the majority of cases (n=242, 90.6%). Septoplasty is an isolated form was carried out only in 25 (9.4%) patients. Results: Good and satisfactory long-term results were noted in 265 (99.2%) cases. An unsatisfactory result occurred in two (0.8%) patients who had a residual deformity of the external nose and deformation of the nasal septum. Various complications after septo- and septorhinoplasty were recorded in 11 (4.1%) cases Conclusion: In most cases (79.4%), the curvature of the nasal septum is combined with the deformation of the external nose of various forms and severity, therefore, the elimination of the curvature of the nasal septum and correction of deformation of the external nose is advisable to perform simultaneously. Preference in performing septorhinoplasty should be given open access, which contributes to a more adequate and correct intervention Keywords: The curvature of the nasal septum, deformity of the external nose, computed tomography, septorhinoplasty, open rhinoplasty.

Higher order Seiberg–Witten functionals and their associated gradient flows

We define functionals generalising the Seiberg–Witten functional on closed $$spin^c$$ manifolds, involving higher order derivatives of the curvature form and spinor field. We then consider their associated gradient flows and, using a gauge fixing technique, are able to prove short time existence for the flows. We then prove energy estimates along the flow, and establish local $$L^2$$ -derivative estimates. These are then used to show long time existence of the flow in sub-critical dimensions. In the critical dimension, we are able to show that long time existence is obstructed by an $$L^{k+2}$$ curvature concentration phenomenon.

Curvature properties of a special type of pure radiation metrics

A spacetime denotes a pure radiation field if its energy–momentum tensor represents a situation in which all the energy is transported in one direction with the speed of light. In 1989 Wils and later in 1997 Ludwig and Edgar studied the physical properties of pure radiation metrics, which are conformally related to a vacuum spacetime. In the present paper we investigate the curvature properties of the pure radiation metrics presented by Ludwig and Edgar. It is shown that such a pure radiation spacetime is semisymmetric, Ricci simple, R-space by Venzi and its Ricci tensor is Riemann compatible. It is also proved that its conformal curvature 2-forms and Ricci 1-forms are recurrent. We also present a pure radiation type metric and evaluate its curvature properties along with the form of its energy–momentum tensor. It is interesting to note that such a pure radiation type metric is Ein(3) and 3-quasi-Einstein. We also find out the sufficient conditions for which this metric represents a generalized pp-wave, pure radiation and perfect fluid. Finally we made a comparison between the curvature properties of Ludwig and Edgar’s pure radiation metric and pp-wave metrics.

Parametric phase transition for a Gauss-Bonnet AdS black hole

With the help of the parametric solution of the Maxwell equal area law for the Gauss-Bonnet AdS black hole in five dimensions, we find the second analytical solution to the first order phase transition. We analyze the asymptotic behaviors of some characteristic thermodynamic properties for the small and large black holes at the critical and zero temperatures and also calculate the critical exponents and the corresponding critical amplitudes in detail. Moreover, we give the general form of the thermodynamic scalar curvature based on the Ruppeiner geometry and point out that the attractive interaction dominates in both the small and large black hole phases when the first order phase transition occurs in the five dimensional Gauss-Bonnet AdS black hole.

A characterization of round spheres in space forms

Let $\mathbb Q^{n+1}_c$ be the complete simply-connected $(n+1)$-dimensional space form of curvature $c$. In this paper we obtain a new characterization of geodesic spheres in $\mathbb Q^{n+1}_c$ in terms of the higher order mean curvatures. In particular, we prove that the geodesic sphere is the only complete bounded immersed hypersurface in $\mathbb Q^{n+1}_c,\;c\leq 0,$ with constant mean curvature and constant scalar curvature. The proof relies on the well known Omori-Yau maximum principle, a formula of Walter for the Laplacian of the $r$-th mean curvature of a hypersurface in a space form, and a classical inequality of G\aa rding for hyperbolic polynomials.

Molecular genetic analysis of genes for detoxification and DNA repair in children with congenital deformities of the thoracic and lumbar spine

Introduction. Spine congenital curvatures, which form from anomalies in the development of vertebral bodies, comprise 3.2% of the general structure of vertebral column deformities. Several such anomalies present during adolescence lead to severe and rigid curvature of the spinal column and are often accompanied by irreversible neurological disorders. The timely detection of the progressive forms of curvature and early surgical treatment are measures that prevent against neurological deficit development and gross congenital deformities of the spine in children. However, it is extremely difficult to predict the course of congenital spinal column deformation in infants based on clinical and radiological investigations alone. Therefore, the study of congenital malformation genetic markers is an essential and immediate task.&#x0D; Materials and methods. Two hundred 1.2–16-year-old children with congenital deformities of the thoracic and lumbar spine were examined using clinical and radiation diagnostic methods. Molecular genetic studies were performed by analyzing several polymorphic regions in the genes for the first and second stages of detoxification and DNA repair, which are of clinical importance as predisposing factors in several congenital malformations. Polymorphisms were determined using the polymerase chain reaction (PCR) method. The results were determined using gel electrophoresis of DNA in a polyacrylamide gel.&#x0D; Results and discussion. The polymorphisms of the genes CYP1A2, NAT2, GSTM1, GSTT1, GSTP1, XRCC1, XRCC3 and their frequency distributions among patients with congenital spine deformities (CSD) were studied. The results for each gene are presented in the digital diagrams, and their indicators are compared with the values of the control group.&#x0D; Conclusion. In most patients (83%) with spinal congenital deformations, there were mutations of candidate genes in the homozygous state; however, the simultaneous carriage of several mutant alleles in patients with CSD was more than twice that in the control group. Children with multiple and combined defects in spine development noted the presence of more mutations in the genes for detoxification and DNA repair. The obtained results already assume to a certain extent the course of the spine congenital deformity in patients at an early age. However, the final evaluation and identification of molecular genetic criteria for the progressive course of spine congenital deformities in children requires further study.

Control of local stress in semiconductor silicon structures

Residual stress distribution in multilayer semiconductor structure is complicated and has a significant impact on device characteristics and yield, therefore their study is one of the actual tasks of modern device engineering. Purpose of the present work was to develop methods of estimation of actual residual stress distribution at the whole area of semiconductor structure and its elements as well.The estimation of residual stress distribution at the area of semiconductor structure was carried out on the basis of determining of local deformation of some areas of the structure by Makyoh topography. This method is based on consequent measurements of intensity of Makyoh image elements of the structure along the chosen direction followed by calculation of micro-geometrical profile and curvature radius.The estimation of residual stress of topological elements Si–SiO system was carried out by means of calculation of interference pictures obtained in a film-substrate gap after separating of film edge from substrate along open window perimeter.Analytical expressions relating semiconductor structure image characteristics with their deformation were developed by means of finite elements method. The expressions allow determining of local residual stress of chosen area of the structure. The examples of stress calculations in real structures are given.Proposed residual stress calculation methods allow to take into consideration character and curvature form of substrate, and also to estimate their magnitude in real topological elements of semiconductor circuits.

Curvature properties of Robinson–Trautman metric

The curvature properties of Robinson–Trautman metric have been investigated. It is shown that Robinson–Trautman metric is a Roter type metric, and in a consequence, admits several kinds of pseudosymmetric type structures such as Weyl pseudosymmetric, Ricci pseudosymmetric, pseudosymmetric Weyl conformal curvature tensor etc. Moreover, it is proved that this metric is a 2-quasi-Einstein, the Ricci tensor is Riemann compatible and its Weyl conformal curvature 2-forms are recurrent. It is also shown that the energy momentum tensor of the metric is pseudosymmetric and the conditions under which such tensor is of Codazzi type and cyclic parallel have been investigated. Finally, we have made a comparison between the curvature properties of Robinson–Trautman metric and Som–Raychaudhuri metric.

Sharp Isoperimetric Inequalities for Small Volumes in Complete Noncompact Riemannian Manifolds of Bounded Geometry Involving the Scalar Curvature

Abstract We provide an isoperimetric comparison theorem for small volumes in an $n$-dimensional Riemannian manifold $(M^n,g)$ with $C^3$ bounded geometry in a suitable sense involving the scalar curvature function. Under $C^3$ bounds of the geometry, if the supremum of scalar curvature function $S_g&amp;lt;n(n-1)k_0$ for some $k_0\in \mathbb{R}$, then for small volumes the isoperimetric profile of $(M^n,g)$ is less then or equal to the isoperimetric profile of the complete simply connected space form of constant sectional curvature $k_0$. This work generalizes Theorem $2$ of [12] in which the same result was proved in the case where $(M^n, g)$ is assumed to be compact. As a consequence of our result we give an asymptotic expansion in Puiseux series up to the 2nd nontrivial term of the isoperimetric profile function for small volumes, generalizing our earlier asymptotic expansion [29]. Finally, as a corollary of our isoperimetric comparison result, it is shown that for small volumes the Aubin–Cartan–Hadamard’s conjecture is true in any dimension $n$ in the special case of manifolds with $C^3$ bounded geometry, and $S_g&amp;lt;n(n-1)k_0$. Two different intrinsic proofs of the fact that an isoperimetric region of small volume is of small diameter. The 1st under the assumption of mild bounded geometry, that is, positive injectivity radius and Ricci curvature bounded below. The 2nd assuming the existence of an upper bound of the sectional curvature, positive injectivity radius, and a lower bound of the Ricci curvature.

Equivariant differential cohomology

The construction of characteristic classes via the curvature form of a connection is one motivation for the refinement of integral cohomology by de Rham cocycles -- known as differential cohomology. We will discuss the analog in the case of a group action on the manifold: The definition of equivariant characteristic forms in the Cartan model due to Nicole Berline and Michele Vergne motivates a refinement of equivariant integral cohomology by all Cartan cocycles. In view of this, we will also review previous definitions critically, in particular the one given by Kiyonori Gomi.

From Hamiltonian to zero curvature formulation for classical integrable boundary conditions

We reconcile the Hamiltonian formalism and the zero curvature representation in the approach to integrable boundary conditions for a classical integrable system in 1 + 1 space-time dimensions. We start from an ultralocal Poisson algebra involving a Lax matrix and two (dynamical) boundary matrices. Sklyanin’s formula for the double-row transfer matrix is used to derive Hamilton’s equations of motion for both the Lax matrix and the boundary matrices in the form of zero curvature equations. A key ingredient of the method is a boundary version of the Semenov-Tian-Shansky formula for the generating function of the time-part of a Lax pair. The procedure is illustrated on the finite Toda chain for which we derive Lax pairs of size for previously known Hamiltonians of type BCN and DN corresponding to constant and dynamical boundary matrices respectively.