ON SHRINKING GRADIENT RICCI SOLITONS WITH POSITIVE RICCI CURVATURE AND SMALL WEYL TENSOR

We show that closed shrinking gradient Ricci solitons with positive Ricci curvature and sufficiently pinched Weyl tensor are Einstein. When Weyl tensor vanishes, this has been proved before but our proof here is much simpler.

Incompatible deformation field and Riemann curvature tensor

Compatibility conditions of a deformation field in continuum mechanics have been revisited via two different routes. One is to use the deformation gradient, and the other is a pure geometric one. Variations of the displacement vector and the displacement density tensor are obtained explicitly in terms of the Riemannian curvature tensor. The explicit relations reconfirm that the compatibility condition is equivalent to the vanishing of the Riemann curvature tensor and reveals the non-Euclidean nature of the space in which the dislocated continuum is imbedded. Comparisons with the theory of Kr¨oner and Le-Stumpf are provided.

M-Eigenvalues of the Riemann Curvature Tensor of Conformally Flat Manifolds

We investigate the M-eigenvalues of the Riemann curvature tensor in the higher dimensional conformally flat manifold.The expressions of Meigenvalues and M-eigenvectors are presented in this paper.As a special case,M-eigenvalues of conformal flat Einstein manifold have also been discussed,and the conformal the invariance of M-eigentriple has been found.We also reveal the relationship between M-eigenvalue and sectional curvature of a Riemannian manifold.We prove that the M-eigenvalue can determine the Riemann curvature tensor uniquely.We also give an example to compute the Meigentriple of de Sitter spacetime which is well-known in general relativity.

Second Order Parallel Tensors on Quasi-constant Curvature Manifolds

The author establishes in this paper the following results: (1) In a quasiconstant curvature manifold M a parallel tensor of type is constant multiple of the metric tensor. (2) On a quasi_constant curvature manifold there is no nonzero parallel 2_form. Unless the Ricci principal curvature corresponding to the generator of M is equal to zero.

Quantum geometric tensor and the topological characterization of the extended Su-Schrieffer-Heeger model

We investigate the quantum metric and topological Euler number in a cyclically modulated Su-Schrieffer-Heeger(SSH)model with long-range hopping terms.By computing the quantum geometry tensor,we derive exact expressions for the quantum metric and Berry curvature of the energy band electrons,and we obtain the phase diagram of the model marked by the first Chern number.Furthermore,we also obtain the topological Euler number of the energy band based on the Gauss-Bonnet theorem on the topological characterization of the closed Bloch states manifold in the first Brillouin zone.However,some regions where the Berry curvature is identically zero in the first Brillouin zone result in the degeneracy of the quantum metric,which leads to ill-defined non-integer topological Euler numbers.Nevertheless,the non-integer"Euler number"provides valuable insights and an upper bound for the absolute values of the Chern numbers.

Curvature range measurements of the arcuate fasciculus using diffusion tensor tractography

Because Broca＇s area and Wernicke＇s area in the brain are connected by the arcuate fasciculus, understanding the anatomical location and morphometry of the arcuate fasciculus can help in the treatment of patients with aphasia. We measured the horizontal and vertical curvature ranges of the arcuate fasciculus in both hemispheres in 12 healthy subjects using diffusion tensor tractography. In the right hemisphere, the direct curvature range and indirect curvature range values of the arcuate fasciculus horizontal part were 121.13 ± 5.89 and 25.99 ± 3.01 degrees, respectively, and in the left hemisphere, the values were 121.83 ± 5.33 and 27.40 ± 2.96 degrees, respectively. In the right hemisphere, the direct curvature range and indirect curvature range values of the arcuate fasciculus vertical part were 43.97 ± 7.98 and 30.15 ± 3.82 degrees, respectively, and in the left hemisphere, the values were 39.39 ± 4.42 and 24.08 ± 4.34 degrees, respectively. We believe that the measured curvature ranges are important data for localization and quantitative assessment of specific neuronal pathways in patients presenting with arcuate fasciculus abnormalities.

Corrigendum:incompatible deformation field and Riemann curvature tensor

The“Corollary 1”formulation in SUN,B.H.Incompatible deformation field and Riemann curvature tensor.Applied Mathematics and Mechanics(English Edition),38(3),311–332(2017)is corrected.It can be stated as follows:The symmetric part of the deformation gradient has no contribution to the trace of the displacement density

Algebraicity of Induced Riemannian Curvature Tensor on Lightlike Warped Product Manifolds

Lightlike warped product manifolds are considered in this paper. The geometry of lightlike submanifolds is difficult to study since the normal vector bundle intersects with the tangent bundle. Due to the degenerate metric, the induced connection is not metric and it follows that the Riemannian curvature tensor is not algebraic. In this situation, some basic techniques of calulus are not useable. In this paper, we consider lightlike warped product as submanifold of semi-Riemannian manifold and establish some remarkable geometric properties from which we establish some conditions on the algebraicity of the induced Riemannian curvature tensor.

WEYL CURVATURE OF A FINSLER SPACE

The Weyl curvature of a Finsler metric is investigated.This curvature constructed from Riemannain curvature.It is an important projective invariant of Finsler metrics.The author gives the necessary conditions on Weyl curvature for a Finsler metric to be Randers metric and presents examples of Randers metrics with non-scalar curvature.A global rigidity theorem for compact Finsler manifolds satisfying such conditions is proved.It is showed that for such a Finsler manifold,if Ricci scalar is negative,then Finsler metric is of Randers type.

Finding Gaussian Curvature of Lifespan Distribution

The objective of this paper is to review the lifespan model. This paper will also suggest four additional general alternative computational methods not mentioned in Kass, R.E. and Vos, P.W. [1] [2]. It is not intended to compare the four formulas to be used in computing the Gaussian curvature. Four different formulas adopted from Struik, D.J. [3] are used and labeled here as (A), (B), (C), and (D). It has been found that all four of these formulas can compute the Gaussian curvature effectively and successfully. To avoid repetition, we only presented results from formulas (B) and (D). One can more easily check other results from formulas (A) and (C).

The Hypersurfaces in a Unit Sphere with Nonnegative Mobius Sectional Curvature

Let x ： M→S^n＋1 be a hypersurface in the （n ＋ 1）-dimensional unit sphere S^n＋1 without umbilic point. The Mobius invariants of x under the Mobius transformation group of S^n＋1 are Mobius metric, Mobius form, Mobius second fundamental form and Blaschke tensor. In this paper, we prove the following theorem： Let x ： M→S^n＋1 （n≥2） be an umbilic free hypersurface in S^n＋1 with nonnegative Mobius sectional curvature and with vanishing Mobius form. Then x is locally Mobius equivalent to one of the following hypersurfaces： （i） the torus S^k（a） × S^n-k（√1- a^2） with 1 ≤ k ≤ n - 1; （ii） the pre-image of the stereographic projection of the standard cylinder S^k × R^n-k belong to R^n＋1 with 1 ≤ k ≤ n- 1; （iii） the pre-image of the stereographic projection of the Cone in R^n＋1 ： -↑x（u, v, t） = （tu, tv）, where （u,v, t）∈S^k（a） × S^n-k-1（ √1-a^2）× R^＋.

SPACE-LIKE BLASCHKE ISOPARAMETRIC SUBMANIFOLDS IN THE LIGHT-CONE OF CONSTANT SCALAR CURVATURE

Let E_(s)^(m+p+1) ?R_(s+1)^(m+p+2)(m≥ 2,p≥ 1,0≤s≤p) be the standard(punched)light-cone in the Lorentzian space R_(s+1)^(m+p+2),and let Y:M^(m)→E_(s)^(m+p+1) be a space-like immersed submanifold of dimension m.Then,in addition to the induced metric g on Mm,there are three other important invariants of Y:the Blaschke tensor A,the conic second fundamental form B,and the conic Mobius form C;these are naturally defined by Y and are all invariant under the group of rigid motions on E_(s)^(m+p+1).In particular,g,A,B,C form a complete invariant system for Y,as was originally shown by C.P.Wang for the case in which s=0.The submanifold Y is said to be Blaschke isoparametric if its conic Mobius form C vanishes identically and all of its Blaschke eigenvalues are constant.In this paper,we study the space-like Blaschke isoparametric submanifolds of a general codimension in the light-cone E_(s)^(m+p+1) for the extremal case in which s=p.We obtain a complete classification theorem for all the m-dimensional space-like Blaschke isoparametric submanifolds in Epm+p+1of constant scalar curvature,and of two distinct Blaschke eigenvalues.

Curvature Energy and Their Spectrum in the Spinor-Twistor Framework: Torsion as Indicium of Gravitational Waves

The twistor kinematic-energy model of the space-time and the kinematic-energy tensor as the energy-matter tensor in relativity are considered to demonstrate the possible behavior of gravity as gravitational waves that derive of mass-energy source in the space-time and whose contorted image is the spectrum of the torsion field acting in the space-time. The energy of this field is the energy of their second curvature. Likewise, it is assumed that the curvature energy as spectral curvature in the twistor kinematic frame is the curvature in twistor-spinor framework, which is the mean result of this work. This demonstrates the lawfulness of the torsion as the indicium of the gravitational waves in the space-time. A censorship to detect gravitational waves in the space-time is designed using the curvature energy.

Space-Time Curvature Mode Quanta

Einstein theorized that Gravity is not a force derived from a potential that acts across a distance. It is a distortion of space and time in which we live by masses and energy. Consistent with Einstein’s theory, a model of space-time curvature modes and associated curvature quanta in slightly warped space-time generated by a light Photon is derived. Both a Schr?dinger and a Second Quantized representation of the space-time curvature mode quanta are calculated and are fourth rank tensors. The eigenvalues of these equations are radii of curvature, not energy. The Eigenfunctions are linear functions of the components of the tensor that describes the curvature of space-time.

A Vector Tensor Calculus Description of a Euclidean Space

We present a tensor description of Euclidean spaces that emphasizes the use of geometric vectors which leads to greater geometric insight and a higher degree of organization in analytical expressions. We demonstrate the effectiveness of the approach by proving a number of integral identities with vector integrands. The presented approach may be aptly described as absolute vector calculus or as vector tensor calculus.

利用重力梯度张量等位面曲率的地质体定位

针对重力梯度张量曲率的研究,前人的工作主要集中在重力张量曲率的解释及边缘检测中,几乎没有涉及到地下密度异常体的定位.本文结合重力矢量和重力梯度张量提出了一套基于重力梯度张量等位面曲率的地下密度异常体位置估计策略.首先,从重力梯度张量等位面曲率的基本定义出发,计算重力梯度张量等位面曲率.然后,通过寻找球面或圆形等位面的重力梯度张量曲率,提出了利用最大主曲率定位地下密度异常体位置的源参数估计方法,并详细推导了估计3D球体(质点)和2D水平线源位置信息的解析表达式.再者,针对噪声和多源存在的情况,提出了一套利用重力梯度张量等位面曲率获得密度异常体位置信息的稳健估计流程,并利用模糊C均值聚类算法进一步确定地下密度异常体的中心位置.最后,通过理论模型测试和文顿盐丘实测航空重力梯度数据测试,验证了本文算法的可行性和可靠性.结果表明:在满足曲率半径定义条件的情况下,本文所提出的源参数估计方法可以定位单个或多个地下3D和2D密度异常体的空间位置,具有较好的稳健性和抗噪能力.该方法拓展了重力梯度张量曲率的应用范围,可为重力梯度张量的三维反演工作提供先验的空间位置信息.

参数空间上的量子几何张量

量子几何张量的实部和虚部均有重要意义,研究二者可以清楚地认识量子系统中的几何与拓扑性质.本文从规范变换作用在实空间上的情况引入,继而延伸到规范变换作用在抽象参数空间上的情况,从而详细地介绍了量子几何张量及一系列概念,加深了对量子几何的进一步理解和认知.

Fourth Rank Energy-Momentum Tensor

In this work, we introduce the new concept of fourth rank energy-momentum tensor. We first show that a fourth rank electromagnetic energy-momentum tensor can be constructed from the second rank electromagnetic energy-momentum tensor. We then generalise to construct a fourth rank stress energy-momentum tensor and apply it to Dirac field of quantum particles. Furthermore, since the established fourth rank energy-momentum tensors have mathematical properties of the Riemann curvature tensor, thus it is reasonable to suggest that quantum fields should also possess geometric structures of a Riemannian manifold.

曲面上的曲率在理论物理中的一些应用

In this survey article,we present two applications of surface curvatures in theoretical physics.The first application arises from biophysics in the study of the shape of cell vesicles involving the minimization of a mean curvature type energy called the Helfrich bending energy.In this formalism,the equilibrium shape of a cell vesicle may present itself in a rich variety of geometric and topological characteristics.We first show that there is an obstruction,arising from the spontaneous curvature,to the existence of a minimizer of the Helfrich energy over the set of embedded ring tori.We then propose a scale-invariant anisotropic bending energy,which extends the Canham energy,and show that it possesses a unique toroidal energy minimizer,up to rescaling,in all parameter regime.Furthermore,we establish some genus-dependent topological lower and upper bounds,which are known to be lacking with the Helfrich energy,for the proposed energy.We also present the shape equation in our context,which extends the Helfrich shape equation.The second application arises from astrophysics in the search for a mechanism for matter accretion in the early universe in the context of cosmic strings.In this formalism,gravitation may simply be stored over a two-surface so that the Einstein tensor is given in terms of the Gauss curvature of the surface which relates itself directly to the Hamiltonian energy density of the matter sector.This setting provides a lucid exhibition of the interplay of the underlying geometry,matter energy,and topological characterization of the system.In both areas of applications,we encounter highly challenging nonlinear partial differential equation problems.We demonstrate that studies on these equations help us to gain understanding of the theoretical physics problems considered.

关于广义Douglas-Weyl喷射的一个注记

该文研究广义D ouglas-Weyl喷射.证明了一个喷射G是广义D ouglas-Weyl喷射当且仅当它的Weyl张量是二次型的.由此得到一个推论,一个芬斯勒度量是广义Douglas-Weyl度量当且仅当它的Weyl张量是二次型的.进一步,该文研究具有二次型的黎曼曲率张量的喷射,证明了一个喷射具有二次型的黎曼曲率张量当且仅当B_(j)^(i)_(kl)=0.