Curvature Properties Research Articles

Three dimensional Sasakian manifolds admitting η-Ricci solitons

In this paper we characterize the three dimensional Sasakian manifolds admitting η-almost Ricci solitons. After the introduction, in section 2, we study three dimensional Sasakian manifolds. In section 3, we prove that an η-Ricci soliton in Sasakian manifolds satisfying the curvature property R.Q = 0 is shrinking and reduces to Ricci soliton. In section 4, we show that the necessary and suficient condition for a Sasakian manifold not admitting a proper η-Ricci soliton is that it is Ricci symmetric. In sections 5 and 6, we study projectively at and concircularly at Sasakian manifold of dimension 3 respectively and find the type of an η-Ricci soliton on such manifold. The next section is devoted to the study of such a manifold admitting η-Ricci soliton and we prove some equivalent conditions. Finally, in section 8, we prove that in a three dimensional Sasakian manifold an η-Ricci soliton becomes Ricci soliton if and only if it is Ricci pseudo-symmetric.

Technical change in U.S. industries

We use the normalized quadratic cost function, introduced by Diewert and Wales (1987), to measure and analyze the rate and biases of technical change at the sectoral level in eleven major U.S. industries — manufacturing, construction, mining, agriculture, finance, health, wholesale, transportation, education, hospitality, and utilities — using annual KLEM (capital, labor, energy, and intermediate materials) data from the World KLEMS database, over the period from 1947 to 2010. We extend the work in Feng and Serletis (2008), by taking a new approach to econometric modeling, merging the econometric approach to productivity measurement with recent state-of-the-art advances in financial econometrics. In particular, we relax the homoskedasticity assumption and instead assume that the covariance matrix of the errors of the flexible interrelated factor demand systems is time-varying. We also pay explicit attention to theoretical regularity, treating the curvature property as a maintained hypothesis, thus achieving superior modeling in the context of a parametric nonlinear factor demand system that captures certain important features of the data.

Projective Changes between Generalized (<i>α</i>, <i>β</i>)-Metric and Randers Metric

Projective change between two Finsler metrics arises from Information Geom-etry. Such metrics have special geometric properties and will play an important role in Finsler geometry. The purpose of the present paper is to find a relation to characterize the projective change between generalized (α, β) - metric ( μ1, μ2 and μ3 ≠ 0 are constants) and Randers metric , where α and are two Riemannian metrics, β and are 1-forms. Further, we study such projective change when generalized (α, β) -metric F has some curvature property.

Statistical connections on decomposable Riemann manifold

Let $(M, g, \varphi)$ be an $n$-dimensional locally decomposable Riemann manifold, that is, $g(\varphi X, Y) = g(X, \varphi Y)$ and $\nabla \varphi = 0$, where $\nabla $ is Riemann (Levi-Civita) connection of metric $g$. In this paper, we construct a new connection on locally decomposable Riemann manifold, whose name is statistical ($\alpha, \varphi)$-connection. A statistical $\alpha $-connection is a torsion-free connection such that $% \overline{\nabla }g = \alpha C$, where $C$ is a completely symmetric $(0, 3)$% -type cubic form. The aim of this article is to use connection $\overline{% \nabla }$ and product structure $\varphi $ in the same equation, which is possible by writing the cubic form $C$ in terms of the product structure $% \varphi $. We examine some curvature properties of the new connection and give examples of it.

Classification of left invariant metrics on 4-dimensional solvable Lie groups

In this paper the complete classification of left invariant metrics of arbitrary signature on solvable Lie groups is given. By identifying the Lie algebra with the algebra of left invariant vector fields on the corresponding Lie group ??, the inner product ??,?? on g = Lie G extends uniquely to a left invariant metric ?? on the Lie group. Therefore, the classification problem is reduced to the problem of classification of pairs (g, ??,??) known as the metric Lie algebras. Although two metric algebras may be isometric even if the corresponding Lie algebras are non-isomorphic, this paper will show that in the 4-dimensional solvable case isometric means isomorphic. Finally, the curvature properties of the obtained metric algebras are considered and, as a corollary, the classification of flat, locally symmetric, Ricciflat, Ricci-parallel and Einstein metrics is also given.

Curvature properties of Kenmotsu manifold admitting semi-symmetric metric connection

Curvature properties of Kenmotsu manifold admitting semi-symmetric metric connection

The Concircular Curvature Tensor Of The Locally Conformal Kahler Manifold

In this research, we are calculated components conharmonic curvature tensor in some aspects Hermeation manifolding in particular of the Locally Conformal Kahler manifold. And we prove that this tensor possesses the classical symmetry properties of the Riemannian curvature. They also, establish relationships between the components of the tensor in this manifold.

Integrable surfaces induced by generalized Landau-Lifshitz equation with self-consistent potential

In this paper, we present integrable surfaces associated with generalized LandauLifshitz equation with self-consistent potential. We obtained the rst and second fundamental forms. The rst fundamental form allow us to calculate the curvature and metric properties of a surface, particularly, length and area of related space. The second fundamental form determines the external geometry of the surface in the vicinity of this point. Together they permit to de ne extrinsic invariants of the surface and its principal curvatures. The results can be used to describe spin waves in magnets and ferromagnets.

Assessment of intraocular pressure with an artificial model eye mimicking human eye

AbstractPurposeGiven that clinical significance of intraocular pressure (IOP), accurate IOP measurements are crucial. However, measurement of IOP is affected by corneal properties such as thickness, curvature, and biomechanical properties. To derive the quantitative relationship between corneal variables and IOP, it is valuable to create a theoretical model that represents these various properties of the cornea. We have constructed optically transparent artificial eye phantoms, which have closely similar to a human eye. The purpose of this study is to evaluate the clinical applicability of artificial eye by understanding and quantifying the IOP measurement.MethodThe artificial model eye has made of polydimethylsiloxane (PDMS) and its inside is filled with glycerol solution. We developed three different shapes of model eye to reflect the variation in corneal diameter and radius of curvature. Corneal properties (i.e., elasticity) were varied according to the formulation of PDMS (Young’s modulus: 1.7 MPa, 0.7 Mpa, and 0.4 MPa) and corneal thickness was differed by varying the concentration of ethylene glycol. The effect of each variable on the measurement of IOP was examined quantitatively using Goldmann applanation tonometry.ResultsThe IOP of the model eye showed significant positive correlation with corneal thickness in all shapes of model eye (all p < 0.001). The range of the IOP changed according to Young’s modulus of model eye. (12–80 mmHg, 1–80 mmHg, and 1–32 mmHg in model eyes with 1.7 MPa, 0.7 MPa, and 0.4 MPa, respectively). Relationship between the IOP and thickness differed depending on the Young’s modulus of the model eye.ConclusionsThe IOP of the developed artificial eyeball was successfully measured by Goldmann applanation tonometer in a similar manner to a real human eye. This finding suggests that the artificial model eye may have a value as an independent, reproducible sample that allows comparison of tonometers.

Kenmotsu Manifolds Admitting a Non-Symmetric Non-Metric Connection

The aim of the present paper is to study the properties of Kenmotsu manifolds equipped with a non-symmetric non-metric connection. We also establish some curvature properties of Kenmotsu manifolds. It is proved that a Kenmotsu manifold endowed with a non-symmetric non-metric is irregular.

A Geometric Strategy Algorithm for Orthogonal Projection onto a Parametric Surface

In this paper, we investigate how to compute the minimum distance between a point and a parametric surface, and then to return the nearest point (foot point) on the surface as well as its corresponding parameter, which is also called the point projection problem of a parametric surface. The geometric strategy algorithm (hereafter GSA) presented consists of two parts as follows. The normal curvature to a given parametric surface is used to find the corresponding foot point firstly, and then the Taylor’s expansion of the parametric surface is employed to compute parameter increments and to get the iteration formula to calculate the orthogonal projection point of test point to the parametric surface. Our geometric strategy algorithm is essentially dependent on the geometric property of the normal curvature, and performs better than existing methods in two ways. Firstly, GSA converges faster than existing methods, such as the method to turn the problem into a root-finding of nonlinear system, subdividing methods, clipping methods, geometric methods (tangent vector and geometric curvature) and hybrid second-order method, etc. Specially, it converges faster than the classical Newton’s iterative method. Secondly, GSA is independent of the initial iterative value, which we prove in Theorem 1. Many numerical examples confirm GSA’s robustness and efficiency.

Curvature Theory on Contact and Transfer Characteristics of Enveloping Curves

Abstract In differential geometry, a curve is characterized by the curvature properties and so is a point trajectory in curvature theory. However, due to the rolling and sliding between contact curves, the characterization of enveloping curves embedded on rigid bodies in relative motion is not complete without the transfer rate of the contact point. This paper presents the new perspectives on not only the curvature characteristics but also the transfer characteristics between enveloping curves embedded on rigid bodies. The paper contains three parts. First, a pair of enveloping curves can be described by a point trajectory which traces two curves simultaneously. Second, the paper treated instant center as a tracing point, which traces the moving centrode and the fixed centrode on each of the bodies. This characterizes the rolling between centrodes and the transfer rate of the instant center on each of centrodes. The treatment can extend the instant center transfer velocity to any higher order. The third part is about the rolling and sliding between enveloping curves embedded on rigid bodies motion. A kinematic model to describe the transfer rate of contact point based on the motion of the instant center is proposed. The detailed curvature and transfer properties of line- and circle-envelopes are presented. Due to the simple treatment, this might be the first paper that successfully achieved the curvature properties of circle-envelopes. The method offers the vital kinematic aspect of enveloping curves distinctly different from the conventional curvature theory and enveloping theory.

Gradient descent optimizes over-parameterized deep ReLU networks

We study the problem of training deep fully connected neural networks with Rectified Linear Unit (ReLU) activation function and cross entropy loss function for binary classification using gradient descent. We show that with proper random weight initialization, gradient descent can find the global minima of the training loss for an over-parameterized deep ReLU network, under certain assumption on the training data. The key idea of our proof is that Gaussian random initialization followed by gradient descent produces a sequence of iterates that stay inside a small perturbation region centered at the initial weights, in which the training loss function of the deep ReLU networks enjoys nice local curvature properties that ensure the global convergence of gradient descent. At the core of our proof technique is (1) a milder assumption on the training data; (2) a sharp analysis of the trajectory length for gradient descent; and (3) a finer characterization of the size of the perturbation region. Compared with the concurrent work (Allen-Zhu et al. in A convergence theory for deep learning via over-parameterization, 2018a; Du et al. in Gradient descent finds global minima of deep neural networks, 2018a) along this line, our result relies on milder over-parameterization condition on the neural network width, and enjoys faster global convergence rate of gradient descent for training deep neural networks.

Study of prestress ultrasonic peening on bending curvature and surface properties of stiffened plates

Stiffened plates are commonly adopted as key structure components in aerospace due to light weight structure with high strength and stiffness. Ultrasonic peening (UP) is the key process for manufacturing of wing panels with stiffened plates; however, large bending strain is difficult to be generated during free UP, which significantly affects its surface properties. Prestress UP is adopted through applying prebending moment to stiffened plates before peening process and its effects on bending curvature, surface hardness and roughness, residual stress distribution under different peening parameters are analyzed. The results show that larger chordwise bending strain can be obtained in prestress UP and it increases with decreasing prebending curvature radius and decreasing offset distance as well as increasing firing pinvelocity. Compared to free UP, smaller surface roughness, larger surface hardness, larger and deeper compressive residual stress can be acquired in prestress UP and their differences between two UP ways increase with decreasing prebending curvature radius. Meanwhile, increasing firing pin velocity and decreasing offset distance are beneficial to hardening extent of peening and generation of large and deep compressive residual stress. However, large firing pin velocity results in large surface roughness and appropriate value of firing pin velocity should be used. Small offset distance will decrease the peening efficiency; therefore, comprehensive consideration of large bending deformation and peening efficiency should be considered. Prestress UP is effective to increase bending strain and improve surface properties of stiffened plate and this study can provide guidance for parameters selection.

Slant Submanifolds of a Lorentz Kenmotsu Manifold

In this paper, we study slant submanifolds of a Lorentz Kenmotsu manifold. Necessary and sufficient conditions are given on a submanifold of a Lorentz Kenmotsu manifold to be a slant submanifold. We also study slant submanifolds of locally warped product Lorentz Kenmotsu manifold. We give examples of slant submanifold warped product a Lorentz Kenmotsu manifold. In addition, we investigate semi-slant submanifolds of a Lorentz Kenmotsu manifold. Moreover, we show that a semi-slant submanifold of locally warped product Lorentz Kenmotsu manifold is a warped product. Furthermore, we obtain some curvature properties for semi-slant submanifold of a Lorentz Kenmotsu space form. Finally, we show that if a semi-slant submanifold of a Lorentz Kenmotsu space form M is totally geodesic, then M is an $$\eta $$ -Einstein manifold.

On Concircular Transformations in Finsler Geometry

A geodesic circle in Finsler geometry is a natural extension of that in a Euclidean space. In this paper, we study geodesic circles and (infinitesimal) concircular transformations on a Finsler manifold. We characterize a concircular vector field with some PDEs on the tangent bundle, and then we obtain respectively necessary and sufficient conditions for a concircular vector field to be conformal and a conformal vector field to be concircular. We also show conditions for two conformally related Finsler metrics to be concircular, and obtain some invariant curvature properties under conformal and concircular transformations.

Classification of left invariant Hermitian structures on 4-dimensional non-compact rank one symmetric spaces

The only 4-dimensional non-compact rank one symmetric spaces are CH2 and RH4. By the classical results of Heintze, one can model these spaces by real solvable Lie groups with left invariant metrics. In this paper we classify all possible left invariant Hermitian structures on these Lie groups, i.e., left invariant Riemannian metrics and the corresponding Hermitian complex structures. We show that each metric from the classification on CH2 admits at least four Hermitian complex structures. One class of metrics on CH2 and all the metrics on RH4 admit 2-spheres of Hermitian complex structures. The standard metric of CH2 is the only Einstein metric from the classification, and also the only metric that admits K¨ahler structure, while on RH4 all the metrics are Einstein. Finally, we examine the geometry of these Lie groups: curvature properties, self-duality, and holonomy.

Finsler warped product metrics with special Riemannian curvature properties

One of the fundamental problems in Finsler geometry is to study Finsler metrics of constant (or scalar) curvature. In this paper, we refine and improve Chen-Shen-Zhao equations characterizing Finsler warped product metrics of scalar flag curvature. In particular, we find equations that characterize Finsler warped product metrics of constant flag curvature. Then we improve the Chen-Shen-Zhao result on characterizing Einstein Finsler warped product metrics. As its application we construct explicitly many new warped product Douglas metrics of constant Ricci curvature by using known locally projectively flat spherically symmetric metrics of constant flag curvature.

Lie groups as 3-dimensional almost paracontact almost paracomplex Riemannian manifolds

Almost paracontact almost paracomplex Riemannian manifolds of the lowest dimension 3 are considered. Such structures are constructed on a family of Lie groups and the obtained manifolds are studied. Curvature properties of these manifolds are investigated. An example is commented as support of obtained results.

A new approach to timelike surfaces which contain inclined curves as geodesics

We study timelike surface, curve pairs by using their curvature properties. We assume that timelike surfaces whose one of the principal curvatures is identically constant, have diagonalizable shape operators in $${\mathbb {E}}_{1}^{3}$$ . Then we give some results about timelike surfaces on which inclined curves lie as geodesic curves.