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.

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.

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.

On the Field Equations of General Relativity

In this work, we examine the geometric character of the field equations of general relativity and propose to formulate relativistic field equations in terms of the Riemann curvature tensor. The resulted relativistic field equations are also integrated into the general framework that we have presented in our previous works that all known classical fields can be expressed in the same dynamical form. We also discuss a possibility to reformulate the field equations of general relativity so that the Ricci curvature tensor and the energy-momentum tensor can appear symmetrically in the field equations without violating the conservation law stated by the covariant derivative.

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 cotan-gent 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 compati-ble paracomplex structures on T＊M. Second, we construct locally decomposable Golden Riemannian structures on T＊M . Finally we investigate curvature properties of T＊M .

Some Interesting Features for External Region of Spherical Symmetric Mass in New Theory of Gravitation VGM

This paper briefly discusses some interesting features for the external region of the spherical symmetric mass in the new theory of gravitation VGM, i.e. the theory of gravitation by considering the vector graviton field and the metric field, such as pseudo-singularity, curvature tensor, static limit, event horizon, and the radial motion of a particle. All these features are different from the corresponding features obtained from general relativity.

A Study on the Second Order Tangent Bundles over Bi-Kählerian Manifolds

This paper aims to study the Berger type deformed Sasaki metric g_(BS)on the second order tangent bundle T^(2)M over a bi-Kählerian manifold M.The authors firstly find the Levi-Civita connection of the Berger type deformed Sasaki metric g_(BS)and calculate all forms of Riemannian curvature tensors of this metric.Also,they study geodesics on the second order tangent bundle T^(2)M and bi-unit second order tangent bundle T^(2)_(1,1)M,and characterize a geodesic of the bi-unit second order tangent bundle in terms of geodesic curvatures of its projection to the base.Finally,they present some conditions for a sectionσ:M→T^(2)M to be harmonic and study the harmonicity of the different canonical projections and inclusions of(T^(2)M,g_(BS)).Moreover,they search the harmonicity of the Berger type deformed Sasaki metric g_(BS)and the Sasaki metric g_(S) with respect to each other.

Complete noncompact manifolds with harmonic curvature

Let （M^n, g） be an n-dimensional complete noncompact Riemannian manifold with harmonic curvature and positive Sobolev constant. In this paper, by employing an elliptic estimation method, we show that （M^n, g） is a space form if it has sufficiently small Ln/2-norms of trace-free curvature tensor and nonnegative scalar curvature. Moreover, we get a gap theorem for （M^n, g） with positive scalar curvature.

Complete bounded λ-hypersurfaces in the weighted volume-preserving mean curvature flow

In this paper, we study the complete bounded λ-hypersurfaces in the weighted volume-preserving mean curvature flow. Firstly, we investigate the volume comparison theorem of complete bounded λ-hypersurfaces with |A|≤α and get some applications of the volume comparison theorem. Secondly, we consider the relation among λ, extrinsic radius k, intrinsic diameter d, and dimension n of the complete λ-hypersurface,and we obtain some estimates for the intrinsic diameter and the extrinsic radius. At last, we get some topological properties of the bounded λ-hypersurface with some natural and general restrictions.

Rigidity Theorems of Riemannian Manifold with 2Ric=0

Ricci curvature tensor is denoted by Ric. We study when the manifold which satisfy 2Ric=0 become a Einstein manifold or a space form.

Some Classes of Kenmotsu Manifolds with Respect to Semi-symmetric Metric Connection

In this paper, we study conharmonic curvature tensor in Kenmotsu manifolds with respect to semi-symmetric metric connection and also characterize conharmonically flat, conharmonically semisymmetric and Ф-conharmonically flat Kenmotsu manifolds with respect to semi-symmetric metric connection.