Curvature Properties Research Articles

Lorentzian stationary surfaces and null curves in $${R^4_2}$$ R 2 4

We discuss a property of the Gaussian curvature and the normal curvature of Lorentzian stationary surfaces in \({R^4_2}\), from the view point of null curves in \({R^4_2}\). We also discuss some examples.

A canonical structure on the tangent bundle of a pseudo- or para-Kähler manifold

It is a classical fact that the cotangent bundle $$T^* {\mathcal {M}}$$ of a differentiable manifold $${\mathcal {M}}$$ enjoys a canonical symplectic form $$\Omega ^*$$ . If $$({\mathcal {M}},\mathrm{J} ,g,\omega )$$ is a pseudo-Kahler or para-Kahler $$2n$$ -dimensional manifold, we prove that the tangent bundle $$T{\mathcal {M}}$$ also enjoys a natural pseudo-Kahler or para-Kahler structure $$({\tilde{\hbox {J}}},\tilde{g},\Omega )$$ , where $$\Omega $$ is the pull-back by $$g$$ of $$\Omega ^*$$ and $$\tilde{g}$$ is a pseudo-Riemannian metric with neutral signature $$(2n,2n)$$ . We investigate the curvature properties of the pair $$({\tilde{\hbox {J}}},\tilde{g})$$ and prove that: $$\tilde{g}$$ is scalar-flat, is not Einstein unless $$g$$ is flat, has nonpositive (resp. nonnegative) Ricci curvature if and only if $$g$$ has nonpositive (resp. nonnegative) Ricci curvature as well, and is locally conformally flat if and only if $$n=1$$ and $$g$$ has constant curvature, or $$n>2$$ and $$g$$ is flat. We also check that (i) the holomorphic sectional curvature of $$({\tilde{\hbox {J}}},\tilde{g})$$ is not constant unless $$g$$ is flat, and (ii) in $$n=1$$ case, that $$\tilde{g}$$ is never anti-self-dual, unless conformally flat.

Some curvature properties of trans-Sasakian manifolds

The object of the present paper is to study quasi-conformally flat trans-Sasakian manifolds. We consider trans-Sasakian manifolds with η-parallel and cyclic parallel Ricci tensors. ϕ-Ricci symmetric quasi-conformally flat trans-Sasakian manifolds have been studied. We also investigates quasi-conformally flat trans-Sasakian manifolds which are Einstein Semi-symmetric.

Riemannian manifolds with two circulant structures

We consider a three-dimensional Riemannian manifold equipped with two circulant structures - a metric g and a structure q, which is an isometry with respect to g and the third power of q is minus identity. We discuss some curvature properties of this manifold, we give an example of such a manifold and find a condition for q to be parallel with respect to the Riemannian connection of g.

Freezing behavior of cement pastes saturated with NaCl solution

This study investigates the freezing behaviors of two cement pastes saturated with water and salt (NaCl) solutions of different concentrations. Special experimental set-up was designed to measure the freezing strains of cylindrical specimens in undrained condition. Using the interfacial curvature properties involved in mercury intrusion under pressure and ice penetration under freezing, the pore ice saturation degree is evaluated through mercury intrusion porosimetry (MIP) data. Experimental results show that both porosity and pore connectivity have impact on the ice saturation degree during freezing. Poromechanical model is established for the freezing strain in the first cooling phase, and the poromechanical simulation agrees reasonably well with the measured strains. The modeling puts in evidence: (1) the freezing strain is induced by pore pressure and thermal contraction of solid matrix, and (2) among the pore pressure contributions the liquid pressure dominates over the interface energy contribution.

Holomorphic curvatures of twistor spaces

We study the twistor spaces of oriented Riemannian 4-manifolds as a source of almost Hermitian 6-manifolds of constant or strictly positive holomorphic, Hermitian and orthogonal bisectional curvatures. In particular, we obtain explicit formulas for these curvatures in the case when the base manifold is Einstein and self-dual, and observe that the "squashed" metric on ℂℙ3 is a non-Kähler Hermitian–Einstein metric of positive holomorphic bisectional curvature. This shows that a recent result of Kalafat and Koca [M. Kalafat and C. Koca, Einstein–Hermitian 4-manifolds of positive bisectional curvature, preprint (2012), arXiv: 1206.3941v1 [math.DG]] in dimension four cannot be extended to higher dimensions. We prove that the Hermitian bisectional curvature of a non-Kähler Hermitian manifold is never a nonzero constant which gives a partial negative answer to a question of Balas and Gauduchon [A. Balas and P. Gauduchon, Any Hermitian metric of constant non-positive (Hermitian) holomorphic sectional curvature on a compact complex surface is Kähler, Math. Z.190 (1985) 39–43]. Finally, motivated by an integrability result of Vezzoni [L. Vezzoni, On the Hermitian curvature of symplectic manifolds, Adv. Geom.7 (2007) 207–214] for almost Kähler manifolds, we study the problem when the holomorphic and the Hermitian bisectional curvatures of an almost Hermitian manifold coincide. We extend the result of Vezzoni to a more general class of almost Hermitian manifolds and describe the twistor spaces having this curvature property.

Research on Algorithm of Gauss Curvature on Space Mesh Points of Battlefield Large-Scale Terrain and its Visualization

This paper discusses the calculation method of Gauss Curvature on space mesh points in simplified level-of-detail (LOD) model technique and how to program with C# language at the platform of .NET. Then, with this algorithm, get the curvature value of the mesh points, delete the center point of the small curvature and triangularize the cavity left so as to realize the simplification of the LOD model. In order to show the result of this algorithm, this paper visualizes the LOD model of the three-dimensional terrain model in full detail with Gauss Curvature Algorithm, puts forward the idea of carrying out error analysis on the algorithm with the geometric property of curvature, and compares the result with that with other algorithms. All the example and the error analysis show that this algorithm is not only of fast speed but also with good effect.

Surface embedding narrow volume reconstruction from unorganized points

In this paper, we present a novel fast and accurate numerical method for the surface embedding narrow volume reconstruction from unorganized points in R3. Though the level set method prevails in the image processing, it requires a redistancing procedure to maintain a desired shape of the level set function. On the other hand, our method is based on the Allen–Cahn equation, which has been applied in image segmentation due to its motion by mean curvature property. We modify the original Allen–Cahn equation by multiplying a control function to restrict the evolution within a narrow band around the given surface data set. To improve the numerical stability of our proposed model, we split the governing equation into linear and nonlinear terms and use an operator splitting technique. The linear equation is solved by the multigrid method which is a fast solver and the nonlinear equation is solved analytically. The unconditional stability of the proposed scheme is also proved. Various numerical results are presented to demonstrate the robustness and accuracy of the proposed method.

The inspection of deformable bodies using curvature estimation and Thompson-Biweight test

Dimensioning and tolerancing standards assume inspection operation, unless otherwise specified, must be done in Free State. This can be problematic when dealing with compliant parts. The inspection of compliant parts needs specialized fixtures because in Free State they may have a significantly different form than their nominal model (CAD) due to inherent variations in the manufacturing process, gravity loads, and residual strains. These specialized fixtures pose difficulties, bear significant costs to industry, and the process is very time-consuming. To address these challenges, this paper proposes a new method for quantifying flexibility/rigidity of the mechanical parts. Subsequently, a novel approach named IDB-CTB is proposed to fixtureless inspection of deformable bodies by curvature estimation and Thompson-Biweight test. This approach combines the Gaussian curvature properties of manufactured compliant parts, one of the intrinsic properties of the geometry, with the Thompson-Biweight statistical test based on the extreme value notion as an identification method. The aim is to distinguish profile deviation due to the manufacturing process from a part’s deformation due to its flexibility in order to determine whether the tolerance fits the CAD model or not. The IDB-CTB approach is tested on two sets of case studies. Three simulated, typical industrial sheet metal case studies were performed in the first set, and an experimental case study in the second one. The low percentage of errors in defect areas and in the profile deviations estimated compared with their reference ones in most cases reflects the effectiveness of the proposed approach.

On the curvature of vortex moduli spaces

We use algebraic topology to investigate local curvature properties of the moduli spaces of gauged vortices on a closed Riemann surface. After computing the homotopy type of the universal cover of the moduli spaces (which are symmetric powers of the surface), we prove that, for genus g>1, the holomorphic bisectional curvature of the vortex metrics cannot always be nonnegative in the multivortex case, and this property extends to all Kaehler metrics on certain symmetric powers. Our result rules out an established and natural conjecture on the geometry of the moduli spaces.

Moser Vector Fields and Geometry of the Mabuchi Moduli Space of Kähler Metrics

There is a natural Moser type transformation along any curve in the moduli spaces of Kähler metrics. In this paper we apply this transformation to give an explicit construction of the parallel transformation along a curve in the Mabuchi moduli space of Kähler metrics. This is crucial in the proof of the equivalence between the existence of the Kähler metrics with constant scalar curvature and the geodesic stability for the type II compact almost homogeneous manifolds of cohomogeneity one mentioned in (Guan 2013). We also explain a new description of the geodesics and prove a curvature property of the moduli space, called curvature symmetric, which makes it similar to some special symmetric spaces with nonpositive curvatures, although the spaces are usually not complete. Finally, we generalize our geodesic stability conjectures in (Guan 2003) and give several results on the Lie algebra structures related to the parallel transformations. In the last section, we generalize the Futaki obstruction of the Kähler-Einstein metrics to the parallel vector fields of the invariant Mabuchi moduli space. We call the related stability the parallel stability. This includes the toric and cohomogeneity one cases as well as the spherical manifolds.

Horizontally submersions of contact $CR$-submanifolds

In this paper, we discuss some geometric properties of almost contact metric submersions involving symplectic manifolds. We show that the structures of quasi-K-cosymplectic and quasi-Kenmotsu manifolds are related to (1, 2)-symplectic structures. For horizontally submersions of contact CR-submanifolds of quasi-K-cosymplectic and quasi-Kenmotsu manifolds, we study the principal characteristics and prove that their total spaces are CR-product. Curvature properties between curvatures of quasi-K-cosymplectic and quasi-Kenmotsu manifolds and the base spaces of such submersions are also established. We finally prove that, under a certain condition, the contact CR-submanifold of a quasi Kenmotsu manifold is locally a product of a totally geodesic leaf of an integrable horizontal distribution and a curve tangent to the normal distribution.

Some curvature properties of Cartan spaces with mth root metrics

In this paper, we define some non-Riemannian curvature properties for Cartan spaces. We consider a Cartan space with the mth root metric. We prove that every mth root Cartan space of isotropic Landsberg curvature, or isotropic mean Landsberg curvature, or isotropic mean Berwald curvature reduces to a Landsberg, weakly Landsberg, and weakly Berwald spaces, respectively. Then we show that the mth root Cartan space of almost vanishing H-curvature satisfies H = 0.

Permeation of Lipidated Protein in Bilayer using Unbiased Simulations Reveals Signature Motif for Protein-Membrane Binding

Structural and dynamic reorganization of membrane domains by proteins is a central theme in many biological processes. Eukaryotic cells possess potential regulatory mechanisms to mediate many of its protein-membrane interactions in form of lipid-modified proteins. Two forms of these proteins exist; cytosolic conformers and functional membrane bound complexes. At present, there are many questions that remain to be answered with regard to the molecular mechanism of interaction of lipidated proteins with membrane that generates a trigger for physiological response, only upon membrane binding.A particularly exciting area of membrane trafficking, autophagy, has been receiving increasing attention. It is a cellular degradative process that involves the formation of autophagosome, a specialized vesicle to deliver the dispensable cellular cargo to the lysosome. During initiation of autophagy, key regulator protein, LC3-I (cytosolic form) is conjugated reversibly to phosphotidylethanolamine (PE) resulting in a nonsoluble form of LC3 that stably associates with the autophagosomal membrane. The present work reports partitioning of PE chain of LC3 in bilayer using MARTINI force field based coarse grain molecular dynamics simulations.Spontaneous insertion of PE chain at microseconds timescale is observed in fourteen out of fifteen trajectories. This unbiased insertion of lipidated protein reveals a novel insertion pathway through an arginine rich patch that drives the insertion of PE chain. Interestingly, membrane curvature properties reveal variations in spontaneous curvature of the membrane, confirming the role of LC3 in forming a vesicle during autophagy. This first unbiased study of active insertion of a lipid chain provides future avenues to investigate detailed regulatory aspects of this phenomenon using experimental techniques.

Local Analysis of Human Cortex in MRI Brain Volume

This paper describes a method for subcortical identification and labeling of 3D medical MRI images. Indeed, the ability to identify similarities between the most characteristic subcortical structures such as sulci and gyri is helpful for human brain mapping studies in general and medical diagnosis in particular. However, these structures vary greatly from one individual to another because they have different geometric properties. For this purpose, we have developed an efficient tool that allows a user to start with brain imaging, to segment the border gray/white matter, to simplify the obtained cortex surface, and to describe this shape locally in order to identify homogeneous features. In this paper, a segmentation procedure using geometric curvature properties that provide an efficient discrimination for local shape is implemented on the brain cortical surface. Experimental results demonstrate the effectiveness and the validity of our approach.

DNA Origami Compliant Nanostructures with Tunable Mechanical Properties

DNA origami enables fabrication of precise nanostructures by programming the self-assembly of DNA. While this approach has been used to make a variety of complex 2D and 3D objects, the mechanical functionality of these structures is limited due to their rigid nature. We explore the fabrication of deformable, or compliant, objects to establish a framework for mechanically functional nanostructures. This compliant design approach is used in macroscopic engineering to make devices including sensors, actuators, and robots. We build compliant nanostructures by utilizing the entropic elasticity of single-stranded DNA (ssDNA) to locally bend bundles of double-stranded DNA into bent geometries whose curvature and mechanical properties can be tuned by controlling the length of ssDNA strands. We demonstrate an ability to achieve a wide range of geometries by adjusting a few strands in the nanostructure design. We further developed a mechanical model to predict both geometry and mechanical properties of our compliant nanostructures that agrees well with experiments. Our results provide a basis for the design of mechanically functional DNA origami devices and materials.

Mass anomaly visualisation and depth estimation from full tensor gradient gravity data

SUMMARY Full Tensor Gradient (FTG) gravity data measures the derivatives of the Earth's gravitational field. Such variations in the gravitational field may be due to the presence of bodies of higher or lower density relative to the surrounding rock. As the gravity tensor contains 5 independent components, effective visualisation of this high-dimensional dataset is advantageous for efficient processing of the FTG data. We present two aspects of visualising mass anomalies in FTG gravity datasets. First, we create a textured image where the orientations of the resulting texture reflect local lateral orientations encoded in the FTG data. It uses a colour map to highlight geologically significant structures such as linear features and radially symmetric points by identifying different geological features and using colour components to represent different feature types. This visualisation method is shown to be robust to significant levels of modelled noise, and we demonstrate its applicability to a field FTG survey. Second, we present an algorithm for estimating the depths of mass anomalies in FTG data. A voxel representation of the subsurface is created and voxels are voted for according to gravitational curvature properties encoded in the FTG tensor. A visualisation of the volume at successive depths highlights 3D locations of mass anomalies at local maxima of the volume. The algorithm is evaluated on a forward-modelled FTG dataset where the depths of mass anomalies are known. The depths of mass anomalies are shown to be accurately located in the presence of noise.

Molecular dynamics simulation of fluid containing gas in hydrophilic rough wall nanochannels

This study performed the molecular dynamic simulations to investigate the boundary behavior of liquid water with entrapped gas bubbles over various hydrophilic roughened substrates. A “liquid–gas–vapor coexistence setup” was employed to maintain a constant thermodynamic state during individual equilibrium simulations and corresponding non-equilibrium Poiseuille flow cases. The two roughened substrates (Si(100) and graphite) adopted in this study present similar contact angles and slip length with gas-free fluid. By considering the effects of argon molecules at the interface, we demonstrated that the boundary slip behavior differed dramatically between these two rough wall channels. This divergence can be attributed to differences in the morphology of argon bubble at the interface due to discrepancies in the atomic arrangement and wall–fluid interaction energy. Furthermore, the density of gas at the interface had a significant impact on the effective slip length of the roughened graphite substrate, whereas shear rate \(\dot{\gamma }\) presented no noticeable influence. On the roughened Si(100) surface, the morphology of the argon bubbles exhibited far higher meniscus curvature and unstable properties under hydrodynamic effects. Thus, this substrate exhibited no slip to slight negative slip and no remarkable influence from either the density of gas at the interface or shear rate. In the present study, we demonstrate that the morphology and behavior of interfacial gas bubbles are influenced by the parameters of wall–fluid interaction as well as the atomic arrangement of the substrate. Our results related to nanochannel flow reveal that different surfaces, such as Si(100) and graphite, may possess similar intrinsic wettability; however, properties of the interfacial gas bubbles can lead to noticeable changes in interfacial characteristics resulting in various degrees of boundary slippage.

Notes on the Rescaled Sasaki Type Metric on the Cotangent Bundle

Let (M,g) be an n-dimensional Riemannian manifold and T*M be its cotangent bundle equipped with the rescaled Sasaki type metric. In this paper, we firstly study the paraholomorphy property of the rescaled Sasaki type metric by using some compatible paracomplex structures on T*M. Second, we construct locally decomposable Golden Riemannian structures on T*M. Finally we investigate curvature properties of T*M.

Analysis of curved beams using a new differential transformation based curved beam element

Differential transformation method is used to obtain the shape functions for nodal variables of an arbitrarily non-uniform curved beam element including the effects of shear deformation considering axially functionally graded material. The proposed shape functions depend on the variations in cross-sectional area, moment of inertia, curvature and material properties along the axis of the curved beam element. The static and free vibration of axially functionally graded tapered curved beams including shear deformation and rotary inertia are studied through solving several examples. Numerical results are presented for circular, parabolic, catenary, elliptic and sinusoidal beams (both forms—prime and quadratic) with hinged-hinged, hinged-clamped and clamped-clamped and clamped-free end restraints. Three general taper types (depth taper, breadth taper and square taper) for rectangular cross section are studied. Out of plane vibration is studied and the lowest natural frequencies are calculated and compared with the published results. Out of plane buckling is investigated for circular beams due to radial load.