The Killing Fields

Heavy metal pollution poses a serious threat to public health Rich in mineral resources, Hunan Province in central China is known as "the land of nonferrous metals.

A Note on Classification of Spatially Homogeneous Rotating Space-Times According to Their Teleparallel Killing Vector Fields in Teleparallel Theory of Gravitation

In this paper we classify spatially homogeneous rotating space-times according to their teleparallel Killing vector fields using direct integration technique.It turns out that the dimension of the teleparallel Killing vector fields is 5 or 10.In the case of 10 teleparallel Killing vector fields the space-time becomes Minkowski and all the torsion components are zero.Teleparallel Killing vector fields in this case are exactly the same as in general relativity.In the cases of 5 teleparallel Killing vector fields we get two more conservation laws in the teleparallel theory of gravitation.Here we also discuss some well-known examples of spatially homogeneous rotating space-times according to their teleparallel Killing vector fields.

Classification of Kantowski-Sachs and Bianchi Type Ⅲ Space-Times According to Their Killing Vector Fields in Teleparallel Theory of Gravitation

In this paper we classify Kantowski-Sachs and Bianchi type Ⅲ space-times according to their teleparallel Killing vector fields using direct integration technique. It turns out that the dimension of the telepaxallel Killing vector fields are 4 or 6, which are the same in numbers as in general relativity. In case of 4 the teleparallel Killing vector fields are multiple of the corresponding Killing vector fields in general relativity by some function of t. In the case of 6 Killing vector fields the metric functions become constants and the Killing vector fields in this case are exactly the same as in general relativity. Here we also discuss the Lie algebra in each case.

Review and Analysis: United States Secret Wars in Cambodia: Long-Term Impacts and Consequences

The People’s Army of Vietnam (PAVN) decided to use conventional warfare to unify Vietnam. The United States military operation against the Ho Chi Minh Trail traffic began in 1959. The (PAVN) connected a series of old trails leading from North Vietnam to South Vietnam via Laos and Cambodia. In the 1960s the network of trails was expanded. Trail traffic was interdicted by repeated CIA (Air America) and US Air Force (Operation Ranch Hand) tactical herbicide spraying and bombing missions. During the late 1960s, the Khmer Rouge Army slowly grew in eastern Cambodia during a time when America was spraying and bombing the Ho Chi Minh Trail. On February 22, 1969, the PAVN launched a new offensive against American forces in South Vietnam from their sanctuaries in Cambodia. President Richard Nixon and Dr. Henry Kissinger, decided to spray and bomb Cambodia, a neutral country, to eliminate the PAVN sanctuary base camps. The damage and loss of life as a result of U.S. air campaign resulted in the insurgency (Khmer Rouge) being able to recruit civilian members. This created a dual effect of strengthening the popularity of the Khmer Rouge, led by Pol Pot, enabling him to overthrow the Khmer Republic in 1975. The primary objectives of this study are to determine the political impacts and consequences of: 1) the 1959 United States secret war on the Ho Chi Minh Trail in Cambodia and 2) the 1969 President Nixon’s decision to destroy the PAVN bases hidden in the Cambodian jungles. These United States secret wars in Cambodia had long-lasting effects on Cambodian political, social, and economic stability.

Generalized elastic curves in the Lorentz flat space L^4

In this paper, we study the extremals of the curvature energy actions on non-null curves in the four-dimensional Lorentz-Minkowski space. We derive the motion equations and find three Killing fields along the generalized elastic curves. We also construct a cylindrical coordinate system using these Killing fields and express the generalized elastic curves by means of quadratures.

The Conformally Symmetric K-Contact Manifolds

In this paper, we establish a differential equation about scalar curvature of conformally flat K-contact manifolds, and prove that a conformally symmetric K-contact manifold is a Riemann manifold with constant curvature 1. At the same time, the results on Sasaki manifolds which are given by Miyazaawa and Yamagushi are generalized to K-contact manifolds.

Generalized Elastica in Anti-de Sitter Space H_1~3

In this paper,the extremals of curvature energy actions on non-null Frenet curves in 3-dimensional Anti-de Sitter space are studied.We completely solve the Euler-Lagrange equation by quadratures.By using the Killing fields,we obtain existence for closed general-ized elastica fully immersed in Anti-de Sitter space H_1~3.

Einstein Finsler metrics and Killing vector fields on Riemannian manifolds

We use a Killing form on a Riemannian manifold to construct a class of Finsler metrics. We find equations that characterize Einstein metrics among this class. In particular, we construct a family of Einstein metrics on S^3 with Ric = 2F^2, Ric = 0 and Ric =-2F^2, respectively. This family of metrics provides an important class of Finsler metrics in dimension three, whose Ricci curvature is a constant, but the flag curvature is not.

BOCHNER TECHNIQUE IN REAL FINSLER MANIFOLDS

Using non-linear connection of Finsler manifold M, the existence of local coordinates which is normalized at a point x is proved, and the Laplace operator A on 1-form of M is defined by non-linear connection and its curvature tensor. After proving the maximum principle theorem of Hopf-Bochner on M, the Bochner type vanishing theorem of Killing vectors and harmonic 1-form are obtained.

Topological Aspects of Entropy and Phase Transition of Kerr Black Holest

In the light of topological current and the relationship between the entropy and the Euler characteristic, the topological aspects of entropy and phase transition of Kerr black holes are studied. From Gauss-Bonnet-Chern theorem, it is shown that the entropy of Kerr black holes is determined by the singularities of the Killing vector field of spacetime. By calculating the Hopf indices and Brouwer degrees of the Killing vector field at the singularities, the entropy S = A/4 for nonextreme Kerr black holes and S = 0 for extreme ones are obtained, respectively. It is also discussed that, with the change of the ratio of mass to angular momentum for unit mass, the Euler characteristic and the entropy of Kerr black holes will change discontinuously when the singularities on Cauchy horizon merge with the singularities on event horizon, which will lead to the first-order phase transition of Kerr black holes.

Euler Characteristic and Topological Phase Transition of NUT-Kerr-Newman Black Hole

From the Gauss-Bonnet-Chern theorem, the Euler characteristic of NUT-Kerr-Newman black hole is calculated to be some discrete numbers from 0 to 2. We find that the Bekenstein-Hawking entropy is the largest entropy in topology by taking into account of the relationship between the entropy and the Euler characteristic. The NUT-Kerr- Newman black hole evolves from the torus-like topological structure to the spherical structure with the changes of mass, angular momentum, electric and NUT charges. In this process, the Euler characteristic and the entropy are changed discontinuously, which give the topological aspect of the first-order phase transition of NUT-Kerr-Newman black hole. The corresponding latent heat of the topological phase transition is also obtained. The estimated latent heat of the black hole evolving from the star just lies in the range of the energy of gamma ray bursts.

Intrinsic Topological Structure of Entropy of Kerr Black Holes

In the light of Ф-mapping method and the relationship between entropy and the Euler characteristic, the intrinsic topological structure of entropy of Kerr black holes is studied. From the Ganss-Bonnet-Chem theorem, it is shown that the entropy of Kerr black hole is determined by singularities of the Killing vector field of spacetime. These singularities naturally carry topological numbers, Hopf indices and Brouwer degrees, which can also be viewed as topological quantization of entropy of Kerr black holes. Specific results S = A/4 for non-extreme Kerr black holes and S = 0 for extreme ones are calculated independently by using the above-mentioned methods.

Generalized Contact Manifolds

We study a Riemannian manifolds M with a f-structures. If the tangent bundle of the Riemannian manifolds is TM=DD⊥ dimD=2P,dimD⊥=q=n-2p≥1. We can define a generalized contact manifolds, get some property of it.

Conformal Vector Fields on Finsler Warped Product Manifolds

In this paper,we study the conformal vector fields on Finsler warped product manifolds.We obtain a system of equivalent equations that the conformal vector fields on Finsler warped product manifolds satisfy and completely characterize conformal vector fields on such manifolds.Further,by solving the equation,we give the classification.And we also give some examples.

A geometric heat flow for vector fields

We introduce and study a geometric heat flow to find Killing vector fields on closed Riemannian manifolds with positive sectional curvature. We study its various properties, prove the global existence of the solution to this flow, discuss its convergence and possible applications, and its relation to the Navier-Stokes equations on manifolds and Kazdan-Warner-Bourguignon-Ezin identity for conformal Killing vector fields. We also provide two new criterions on the existence of Killing vector fields. A similar flow to finding holomorphic vector fields on K¨ahler manifolds will be studied by Li and Liu(2014).

A Kenmotsu Metric as a *-conformal Yamabe Soliton with Torse Forming Potential Vector Field

The purpose of the present paper is to deliberate *-conformal Yamabe soliton,whose potential vector field is torse-forming on Kenmotsu manifold.Here,we have shown the nature of the soliton and find the scalar curvature when the manifold admitting *-conformal Yamabe soliton on Kenmotsu manifold.Next,we have evolved the characterization of the vector field when the manifold satisfies *-conformal Yamabe soliton.Also we have embellished some applications of vector field as torse-forming in terms of *-conformal Yamabe soliton on Kenmotsu manifold.We have developed an example of *-conformal Yamabe soliton on 3-dimensional Kenmotsu manifold to prove our findings.

Homogeneous Finsler spaces with only one orbit of prime closed geodesics

When a closed Finsler manifold admits continuous isometric actions,estimating the number of orbits of prime closed geodesics seems a more reasonable substitution for estimating the number of prime closed geodesics.To extend the results of Duan,Long,Rademacher,Wang and others on the existence of two prime closed geodesics to the equivariant situation,we propose the question if a closed Finsler manifold has only one orbit of prime closed geodesics if and only if it is a compact rank-one Riemannian symmetric space.In this paper,we study this problem in homogeneous Finsler geometry,and get a positive answer when the dimension is even or the metric is reversible.We guess the rank inequality and the algebraic techniques in this paper may continue to play an important role for discussing our question in the non-homogeneous situation.

Geometry of the Second-Order Tangent Bundles of Riemannian Manifolds

Let （M, g） be an n-dimensional Riemannian manifold and T2M be its second- order tangent bundle equipped with a lift metric g. In this paper, first, the authors con- struct some Riemannian almost product structures on （T2M, g） and present some results concerning these structures. Then, they investigate the curvature properties of （T2M, g）. Finally, they study the properties of two metric connections with nonvanishing torsion on （T2M, g： The//-lift of the Levi-Civita connection of g to TaM, and the product conjugate connection defined by the Levi-Civita connection of g and an almost product structure.