COMPLETE KAHLER METRICS WITH POSITIVE HOLOMORPHIC SECTIONAL CURVATURES ON CERTAIN LINE BUNDLES(RELATED TO A COHOMOGENEITY ONE POINT OF VIEW ON A YAU CONJECTURE)

In this article,we study Kahler metrics on a certain line bundle over some compact Kahler manifolds to find complete Kahler metrics with positive holomorphic sectional(or bisectional)curvatures.Thus,we apply a strategy to a famous Yau conjecture with a co-homogeneity one geometry.

Distribution and correlation of refractive parameters in children with different corneal curvatures in southeast China

AIM:To analyze the distribution of refractive status in school-age children with different corneal curvatures(CC)and the correlation between CC and refractive status.METHODS:A total of 2214 school-aged children of grade 4 in Hangzhou who were screened for school myopia were included.Uncorrected distance visual acuity(UCDVA),non-cycloplegic refraction,axial length(AL),horizontal and vertical corneal curvature(K1,K2)were measured and spherical equivalent(SE),corneal curvature radius(CCR)and axial length/corneal radius of curvature ratio(AL/CR)were calculated.UCDVA<5.0 and SE≤-0.50 D were classified as school-screening myopia.According to the different CCRs,the patients were divided into the lower corneal curvature(LCC)group(CCR≥7.92)and the higher corneal curvature(HCC)group(CCR<7.92).Each group was further divided into the normal AL subgroup and the long AL subgroup.The refractive parameters were compared to identify any differences between the two groups.RESULTS:Both SE and AL were greater in the LCC group(P=0.013,P<0.001).The prevalence of myopia was 38% in the LCC group and 44% in the HCC group(P<0.001).The proportion of children without screening myopia was higher in the LCC group(62%)than in the HCC group(56%).Among these children without screening myopia,the proportion of long AL in the LCC group(24%)was significantly higher than that in the HCC group(0.012%;P<0.001).The change of SE in the LCC group was less affected by the increase of AL than that in the HCC group.CONCLUSION:School-aged children in the LCC group have a lower incidence of screening myopia and longer AL.Low CC can mask SE reduction and AL growth to some extent,and the change of AL growth change more in children with low CC than high CC.Before the onset of myopia,its growth rate is even faster than that after the onset of myopia.

Curvatures estimation on triangular mesh

Curvatures are important geometric attributes of surfaces. There are many applications that require as a first step the accurate estimation of curvatures at arbitrary vertices on a triangulated surface. Chen and Schmitt (1992) and Taubin (1995) presented two simple methods to estimate principal curvatures. They used circular arcs to approximate the normal curvature. We find this may cause large error in some cases. In this paper, we describe a more accurate method to estimate the normal curvature, and present a novel algorithm to estimate principal curvatures by simplifying the Chen and Schmitt’s method. Some comparison results are also shown in this paper.

Quantum Gravity, Constant Negative Curvatures, and Black Holes

For purposes of quantization, classical gravity is normally expressed by canonical variables, namely the metric and the momentum . Canonical quantization requires a proper promotion of these classical variables to quantum operators, which, according to Dirac, the favored operators should be those arising from classical variables that formed Cartesian coordinates;sadly, in this case, that is not possible. However, an affine quantization features promoting the metric and the momentric to operators. Instead of these classical variables belonging to a constant zero curvature space (i.e., instead of a flat space), they belong to a space of constant negative curvatures. This feature may even have its appearance in black holes, which could strongly point toward an affine quantization approach to quantize gravity.

Non-uniform,axisymmetric misfit strain:in thin films bonded on plate substrates/substrate systems:the relation between non-uniform film stresses and system curvatures

Current methodologies used for the inference of thin film stress through curvature measurements are strictly restricted to stress and curvature states which are assumed to remain uniform over the entire film/substrate system. By considering a circular thin film/substrate system subject to non-uniform, but axisymmetric misfit strain distributions in the thin film, we derived relations between the film stresses and the misfit strain, and between the plate system＇s curvatures and the misfit strain. These relations feature a “local” part which involves a direct dependence of the stress or curvature components on the misfit strain at the same point, and a “non-local” part which reflects the effect of misfit strain of other points on the location of scrutiny. Most notably, we also derived relations between the polar components of the film stress and those of system curvatures which allow for the experimental inference of such stresses from full-field curvature measurements in the presence of arbitrary radial non-uniformities. These relations also feature a “non-local” dependence on curvatures making a full-field measurement a necessity. Finally, it is shown that the interfacial shear tractions between the film and the substrate are proportional to the radial gradients of the first curvature invariant and can also be inferred experimentally.

ON MEAN CURVATURES OF A PARALLEL CONVEX BODY

In this article, we obtain some results about the mean curvature integrals of the parallel body of a convex set in R^n. These mean curvature integrals are generalizations of the Santalo＇s results.

On Fibre Bundle Surfaces and Their Curvatures

A surface model called the fibre bundle model is proposed. This model represents a surface locally as a direct product of two curves： a base curve and a fibre curve. We introduce the fibre bundle model and then obtain the Gaussian curvatures and the mean curvatures of a certain kind of fibre bundle surface models using 1-parameter groups of a linear Lie algebra as fibres. Some examples are given to verify our results.

Interactions of human islet amyloid polypeptide with lipid structure of different curvatures

Curvature is one of the most important features of lipid membranes in living cells,which significantly influences the structure of lipid membranes and their interaction with proteins.Taken the human islet amyloid polypeptide(h IAPP),an important protein related to the pathogenesis of type II diabetes,as an example,we performed molecular dynamics(MD)simulations to study the interaction between the protein and the lipid structures with varied curvatures.We found that the lipids in the high curvature membrane pack loosely with high mobility.The h IAPP initially forms H-bonds with the membrane surface that anchored the protein,and then inserts into the membrane through the hydrophobic interactions between the residues and the hydrophobic tails of the lipids.h IAPP can insert into the membrane more deeply with a larger curvature and with a stronger binding strength.Our result provided important insights into the mechanism of the membrane curvature-dependent property of proteins with molecular details.

Shape gradient and classical gradient of curvatures:driving forces on micro/nano curved surfaces

Recent experiments and molecule dynamics simulations have shown that adhesion droplets on conical surfaces may move spontaneously and directionally. Besides, this spontaneous and directional motion is independent of the hydrophilicity and hydrophobicity of the conical surfaces. Aimed at this important phenomenon, a gen- eral theoretical explanation is provided from the viewpoint of the geometrization of micro/nano mechanics on curved surfaces. In the extrinsic mechanics on micro/nano soft curved surfaces, we disclose that the curvatures and their extrinsic gradients form the driving forces on the curved spaces. This paper focuses on the intrinsic mechanics on micro/nano hard curved surfaces and the experiment on the spontaneous and directional motion. Based on the pair potentials of particles, the interactions between an isolated particle and a micro/nano hard curved surface are studied, and the geometric foundation for the interactions between the particle and the hard curved surface is analyzed. The following results are derived： （a） Whatever the exponents in the pair potentials may be, the potential of the particle/hard curved surface is always of the unified curvature form, i.e., the potential is always a unified function of the mean curvature and the Gaussian curvature of the curved surface. （b） On the basis of the curvature-based potential, the geometrization of the micro/nano mechanics on hard curved surfaces may be realized. （c） Similar to the extrinsic mechanics on micro/nano soft curved surfaces, in the intrinsic mechanics on micro/nano hard curved surfaces, the curvatures and their intrinsic gradi- ents form the driving forces on the curved spaces. In other words, either on soft curved surfaces or hard curved surfaces and either in the extrinsic mechanics or the intrinsic mechanics, the curvatures and their gradients are all essential factors for the driving forces on the curved spaces. （d） The direction of the driving force induced by the hard curved surface is independent of the hydrophilieity and hydrophobicity of the curved surface, explaining the experimental phenomenon of the spontaneous and directional motion.

Reconstruction of Three Dimensional Convex Bodies from the Curvatures of Their Shadows

In this article, we study necessary and sufficient conditions for a function, defined on the space of flags to be the projection curvature radius function for a convex body. This type of inverse problems has been studied by Christoffel, Minkwoski for the case of mean and Gauss curvatures. We suggest an algorithm of reconstruction of a convex body from its projection curvature radius function by finding a representation for the support function of the body. We lead the problem to a system of differential equations of second order on the sphere and solve it applying a consistency method suggested by the author of the article.

Riparian plant community structure and assembly processes differed by variations in riverbank curvatures:implications for restoring habitats along the Three Gorges Reservoir

A meandering riverbank plays a vital role in maintaining natural river ecosystems,providing habitats for riparian vegetation.However,dams have significantly altered riverbank shapes.To restore the riparian ecosystems,it is imperative to understand how different riverbank curvatures influence them.This study aims to uncover the ecological impacts of riverbank curvature on the structure and assembly process of plant communities in the riparian zone of the Yangtze River,regulated by the Three Gorges Dam(TGD)in China.We categorized the riparian zones into four types:cove,lobe,wavy and linear shapes.We documented the composition and diversity of riparian plant communities.Our findings revealed that wavy and cove riverbanks exhibited greater species diversity(with Shannon–Wiener diversity index values 1.5×higher)compared to communities along linear riverbanks.Furthermore,the analysis of functional traits indicated that wavy riverbanks promoted the differentiation of plant functional traits,thus enhancing ecosystem functions,with functional dispersion index(FDis)values 1.3 times higher than those of linear riverbanks.Significant variations in the assembly of riparian communities were also observed among different riverbanks,with standardized effect size(SES)values indicating a higher degree of niche differentiation in cove riverbanks(SES=0.4)compared to linear riverbanks(SES=–0.6).These results highlight the ecological importance of diverse riverbank curvatures in influencing the diversity,structure and assembly of riparian communities along the waterway.In summary,this study underscores the necessity of maintaining or restoring various natural morphological curvatures when rehabilitating riparian communities along rivers impacted by human activities.

CLASSIFICATIONS OF DUPIN HYPERSURFACES IN LIE SPHERE GEOMETRY

This is a survey of local and global classification results concerning Dupin hypersurfaces in S^(n)(or R^(n))that have been obtained in the context of Lie sphere geometry.The emphasis is on results that relate Dupin hypersurfaces to isoparametric hypersurfaces in spheres.Along with these classification results,many important concepts from Lie sphere geometry,such as curvature spheres,Lie curvatures,and Legendre lifts of submanifolds of S^(n)(or R^(n)),are described in detail.The paper also contains several important constructions of Dupin hypersurfaces with certain special properties.

On the Negativity of Ricci Curvatures of Complete Conformal Metrics

A version of the singular Yamabe problem in bounded domains yields complete conformal metrics with negative constant scalar curvatures.In this paper,we study whether these metrics have negative Ricci curvatures.Affirmatively,we prove that these metrics indeed have negative Ricci curvatures in bounded convex domains in the Euclidean space.On the other hand,we provide a general construction of domains in compact manifolds and demonstrate that the negativity of Ricci curvatures does not hold if the boundary is close to certain sets of low dimension.The expansion of the Green’s function and the positive mass theorem play essential roles in certain cases.

Design and Optimization of Press Bend Forming Path for Producing Aircraft Integral Panels with Compound Curvatures

In order to find out the optimal press bend forming path in fabricating aircraft integral panels, this article proposes a new method on the basis of the authors＇ previous work. It is composed of the finite element method （FEM） equivalent model, the surface curvature analysis, the artificial neural network response surface and the genetic algorithm. The method begins with analyzing the objective＇s shape curvature to determine the bending position. Then it optimizes the punch travel at each bending position by the following steps： （1） Establish a multi-step press bend forming FEM equivalent model, with which the FEM ex- periments designed with the Taguchi method are performed. （2） Construct a back-propagation （BP） neural network response surface with the data from the FEM experiments. （3） Use the genetic algorithm to optimize the neural network response surface as the objective function. Finally, this method is verified by press bending a complicated double-curvature grid-type stiffened panel and bears out its effectiveness and intrinsic worth in designing the press bend forming path.

Interaction potential between micro/nano curved surface and a particle located inside the surface (I):driving forces induced by curvatures

This paper focuses on the interaction between a micro/nano curved surface and a particle located inside the surface (hereafter abbreviated as in-surface-particle).Based on the exponential pair potential (namely 1/R2k) between particles,the interaction potential between the micro/nano curved surface and the in-surface-particle is derived.The following results are shown:(a) For an even number of exponents in the pair potential,the interaction potential between the micro/nano curved surface and the in-surface-particle can be expressed as a unified function of the mean curvature and Gaussian curvature of the curved surface;(b) the curvatures and the gradients of curvatures of the micro/nano curved surface are the essential factors that dominate the driving force acting on the particle.

Chern-Ricci curvatures, holomorphic sectional curvature and Hermitian metrics

We present some formulae related to the Chern-Ricci curvatures and scalar curvatures of special Hermitian metrics.We prove that a compact locally conformal Kähler manifold with the constant nonpositive holomorphic sectional curvature is K?hler.We also give examples of complete non-Kähler metrics with pointwise negative constant but not globally constant holomorphic sectional curvature,and complete non-Kähler metrics with zero holomorphic sectional curvature and nonvanishing curvature tensors.

A STABILITY RESULT FOR TRANSLATINGSPACELIKE GRAPHS IN LORENTZ MANIFOLDS

In this paper,we investigate spacelike graphs defined over a domain Ω⊂M^(n) in the Lorentz manifold M^(n)×ℝ with the metric−ds^(2)+σ,where M^(n) is a complete Riemannian n-manifold with the metricσ,Ωhas piecewise smooth boundary,and ℝ denotes the Euclidean 1-space.We prove an interesting stability result for translating spacelike graphs in M^(n)×ℝ under a conformal transformation.

Laguerre Isoparametric Hypersurfaces in R^n with Two Distinct Non-zero Principal Curvatures

An umbilical free oriented hypersurfacex：M→Rnwith non-zero principal curvatures is called a Laguerre isoparametric hypersurface if its Laguerre form C=i Ciωi=iρ1（Ei（logρ）（r ri）Ei（r））ωi vanishes and Laguerre shape operator S=ρ1（S 1 rid）has constant eigenvalues.Hereρ=i（r ri）2,r=r1＋r2＋···＋rn 1n 1is the mean curvature radius andSis the shape operator ofx.{Ei}is a local basis for Laguerre metric g=ρ2III with dual basis{ωi}and III is the third fundamental form ofx.In this paper,we classify all Laguerre isoparametric hypersurfaces in Rn（n〉3）with two distinct non-zero principal curvatures up to Laguerre transformations.

Nonholonomic Theory of Principal-direction Orthonormal Basis for a Layer of Surfaces

In order to carry out tensor analysis in a neighborhood of a reference surface,the principal-direction orthogonal basis accompanying with Lame s coefficients or general curvilinear coordinate systems are widely used.A novel kind of field theory termed as the nonholonomic theory of the Principal-Direction Orthonormal Basis(PDOB)is presented systematically in the present paper,in which the formal Christoffel symbols are related directly to the principal and geodesic curvatures with respect to the principal directions of the surface.Furthermore,a systematic and simple way to determine the curvatures of the surface are presented with some examples.It provides a way to recognize qualitatively the bending property of a surface.

Layer-Valley Hall Effect under Inversion and Time-Reversal Symmetries

Hall effects have been the central paradigms in modern physics,materials science and practical applications,and have led to many exciting breakthroughs,including the discovery of topological Chern invariants and the revolution of metrological resistance standard.To date,the Hall effects have mainly focused on a single degree of freedom(Do F),and most of them require the breaking of spatial-inversion and/or time-reversal symmetries.Here we demonstrate a new type of Hall effect,i.e.,layer-valley Hall effect,based on a combined layer-valley Do F characterized by the product of layer and valley indices.The layer-valley Hall effect has a quantum origin arising from the layer-valley contrasting Berry curvature,and can occur in nonmagnetic centrosymmetric crystals with both spatial-inversion and time-reversal symmetries,transcending the symmetry constraints of single Do F Hall effect based on the constituent layer or valley index.Moreover,the layer-valley Hall effect is highly tunable and shows a W-shaped pattern in response to the out-of-plane electric fields.Additionally,we discuss the potential detection approaches and material-specific design principles of layer-valley Hall effect.Our results demonstrate novel Hall physics and open up exotic paradigms for new research direction of layer-valleytronics that exploits the quantum nature of the coupled layer-valley DoF.