THE HOMOTHETIC MOTIONS IN THE LORENTZ 3-SPACE

In this article, the properties of the homothetic motions in three-dimensional Lorentz space are investigated. Also, some geometric results between velocity and acceleration vectors of a point in a spatial motion are obtained.

HAMILTON OPERATORS AND HOMOTHETIC MOTIONS IN R^3

Quaternion is a division ring.It is shown that planes passing through the origin can be made a field with the quaternion product in R~3.The Hamiltonian operators help us define the homothetic motions on these planes.New characterizations for these motions are investigated.

GENERALIZATION OF STEINER FORMULAFOR THE HOMOTHETIC MOTIONS ONTHE PLANAR KINEMATICS

The Steiner formula and the mixture area formula given by M(U|¨)ller were expressed under the one-parameter closed planar homothetic motions in the complex sense . Also, using the generalization of Steiner formula, the result of Holditch theorem for homothetic motions is got. In the case of the homothetic scale h≡1 the results given by M(U|¨)ller are obtained as a special case.

Local Study of Scalar Curvature of Cyclic Surfaces Obtained by Homothetic Motion of Lorentzian Circle

In this paper we consider the homothetic motion of Lorentzian circle by studying the scalar curvature for the corresponding cyclic surface locally. We prove that if the scalar curvature is constant, then . We describe the equations that govern such surfaces.

On the Kinematics for the Closed Planar Homothetic Inverse Motions in Complex Plane

In this paper, the Steiner area formula and the polar moment of inertia were expressed during one-parameter closed planar homothetic inverse motions in complex plane. The Steiner point was defined when the rotation number was different zero and it was called the Steiner normal when the rotation number was equal to zero. The fixed pole point was given with its components and its relation between Steiner point or Steiner normal was explained. The sagittal motion of a telescopic crane was considered as an example. This motion was described by a double hinge consisting of the fixed control panel of the telescopic crane and the moving arm of the telescopic crane. The theoretical concepts and results were applied for this motion.

On the Kinetic Energy of the Projection Curve for the 1-Parameter Closed Spatial Homothetic Motion

In this study, the kinetic energy formula of the projection curve under 1-parameter closed homothetic motion is expressed and as a result, theorem is given. Also some special cases are given related with that formula.

The Formula of Kinetic Energy for the Closed Planar Homothetic Direct Motions

In this paper, during one-parameter closed planar homothetic direct motions, the formula of kinetic energy is expressed. Then we show the relation between the formula of kinetic energy and the Steiner formula. We investigate some properties of closed planar homothetic motions. These motions appear between two coordinate systems, fixed and moving (direct motion). Finally, we show how the results can be applied to experimentally measured motions. As an example, we consider a motion of winch in the sagittal direction. We obtain the formula of kinetic energy for the motion of winch during one-parameter closed planar homothetic direct motions.

The Kinetic Energy Formula for the Closed Planar Homothetic Inverse Motions in Complex Plane

In this paper, the kinetic energy formula was expressed during one-parameter closed planar homothetic inverse motions in complex plane. Then the relation between the kinetic energy formula and the Steiner formula was given. As an example the sagittal motion of a telescopic crane was considered. This motion was described by a double hinge consisting of the fixed control panel of telescopic crane and the moving arm of telescopic crane. The results were applied to experimentally measured motion.

Proper Teleparallel Homothetic Vector Fields in General Cylindrically Symmetric Space-Times in Teleparallel Theory of Gravitation Using Diagonal Tetrads

In this paper we consider the most general form of non-static cylindrically symmetric space-times in order to study proper teleparallel homothetic vector fields using the direct integration technique and diagonal tetrads. This study also covers static cylindrically symmetric, Bianchi type I, non-static and static plane symmetric space-times as well. Here, we will only discuss the cases which do not fall in the category of static cylindrically symmetric, Bianchi type Ⅰ, non-static and static plane symmetric space-times. From the above study we show that very special classes of the above space-times yield 6, 7 and 8 teleparallel homothetic vector fields with non-zero torsion.

ON(α,β)-METRICS OF CONSTANT FLAG CURVATURE

In this paper,we study the(α,β)-metrics of constant flag curvature.We characterize almost regular(α,β)-metrics of constant flag curvature under the condition that β is a homothetic 1-form with respect to a.Furthermore,we prove that if a regular(α,β)-metric is of constant flag curvature and β is a Killing 1-form with constant length,then it must be a Riemannian metric or locally Minkowskian.

Self-Similar Solutions of the Kantowski-Sachs Model with a Perfect Fluid in General Relativity

Exact self-similar solutions to Einstein’s field equations for the Kantowski-Sachs space-time are determined. The self-similarity property is applied to determine the functional form of the unknown functions that define the gravitational model and to reduce the order of the field equations. The consequences of matter, described by the energy-momentum tensor, are investigated in the case of a perfect fluid. Some physical features and kinematical properties of the obtained model are studied.

Isoparametric Hypersurfaces Induced by Navigation in Lorentz Finsler Geometry

Using a navigation process with the datum(F,V),in which F is a Finsler metric and the smooth tangent vector field V satisfies F(−V(x))>1 everywhere,a Lorentz Finsler metric F˜can be induced.Isoparametric functions and isoparametric hypersurfaces with or without involving a smooth measure can be defined for F˜.When the vector field V in the navigation datum is homothetic,we prove the local correspondences between isoparametric functions and isoparametric hypersurfaces before and after this navigation process.Using these correspondences,we provide some examples of isoparametric functions and isoparametric hypersurfaces on a Funk space of Lorentz Randers type.

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.