"HyperComplex Numbers in Geometry and Physics" 2 (4), vol. 2, 2005
j004

Content of Issue is in the theme. The journal in one file is below.

2005jbz

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2005jby

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2005jbx | Pavlov D. G.

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2005jbw

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2005jbv

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The generalized Finslerian metric tensors
2005jbu | Lebedev S. V.  // Baumann University's Institute of Applied Math@Mech

The generalized Finslerian metric tensors are proposed. These tensors can have different number of indeces dependent on space dimension as well as space properties. The relationship of these tensors with the Finsler spaces associated with commutative associative algebras is analyzed. Nearest perspectives to research of the tensors of this type are discussed. The generalized differential equations of Finsler geodesics are derived and discussed.

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2005jbt

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2005jbs

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Finsler spinors
2005jbr

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The prolongations of a Finsler metric to the tangent bunde $T^k(M) (k>1)$ of the higher order accelerations
2005jbq | Atanasiu Gh.  // Department of Algebra and Geometry, Transilvania University, Brasov, Romania

An old problem in differential geometry is that of prolongation of a Riemannian structure $g\left( x\right) $ on a real $n-$dimensional $% C^{\infty }$-manifold $M,x\in M,$ to the bundle of $k-$jets $\left( J_{0}^{k}M,\pi ^{k},M\right) $ or, equivalently the tangent bundle $\left( T^{k}M,\pi ^{k},M\right) $ of the higher order accelerations. The problem belongs to so-called geometry of higher order. It was solved in $\left[ 18% \right] $ for $k=1$ and partially in $\left[ 19\right] $ for$\;k=2.$ The same problem of prolongation can be considered for a Finslerian structure $% F\left( x,y^{\left( 1\right) }\right) $. In the paper $\left[ 15\right] $ are given these solutions in the general cases, using the Sasaki-Matsumoto $N-$lift (for $k=2,$ see $\left[ 3\right] $ and $\left[ 6\right] ).

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The 2-Cotangent Bundle with Berwald-Moor Metric
2005jbp | Gheorghe Atanasiu, Vladimir Balan  // Transilvania University, Brasov, Romania; University Politehnica of Bucharest, Department Mathematics I, Romania

On the total space of the dual bundle $(T^{\ast 2}M,\pi ^{\ast 2},M)$ of the $2-$tangent bundle $(T^{2}M, \pi ^{2},M)$, the paper develops results related to the notions: of nonlinear connection, distinguished tensor fields, almost contact structure, Riemannian structures, $N-$linear connections and associated convariant derivations. The Ricci identities are derived and the local expressions of the corresponding $d-$tensors of torsion and curvature are provided. Further, the metric structures and the metric $N-$linear connections are studied, and the obtained results are specialized to the case when the metric tensor field is of Berwald-Moor type.

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The Berwald-Moor metric in the tangent bundle of the second order
2005jbo | Gheorghe Atanasiu, Nicoleta Brinzei

As an application of the results of the first author obtained in the papers \cite{1} and \cite{2}, the geometry of the second order tangent bundle $% T^{2}M$ (or second order jet bundle $J_{0}^{2}M$) endowed with two special types of metrics compatible with the 2-contact structures is studied. The particularity of these two models is that the horizontal and the $v^{(1)}$-\ part of the metric are both given by the same Riemannian metric (respectively, its horizontal part is Riemannian), while its $v^{(2)}$-part is given by the flag-Finsler Berwald-Moor metric (respectively, the $v^{(1)} $ and $v^{(2)}$- parts are given by the flag-Finsler Berwald-Moor metric, \cite{Mangalia}).

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Berwald-Moor-type $(h,v)$-metric physical models
2005jbn | Balan V., Brinzei N.  // University Politehnica of Bucharest, Department Mathematics I; Department of Mathematics, "Transilvania" University, Brasov, Romania

In the framework of vector bundles endowed with $(h,v)-$metrics several physical models for relativity are presented. A characteristic of these models is that the vertical part is provided by the flag-Finsler Berwald-Moor (fFBM) metric, while the horizontal part is specialized to the conformal and to Synge-relativistic optics metrics. As well, the particular case of $h-$Riemannian $v-$fFBM metric of Riemann-Minkowski type is examined, considering as nonlinear connection both the trivial canonical connection, and the one induced by the Lagrangian of electrodynamics. For all these models, basic properties are described and the extended Einstein and Maxwell equations are determined.

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Invariant frames for a generalized Lagrange space with Berwald-Moor metric
2005jbm | Marius Paun  // Faculty of Mathematics and Informatics, Transilvania, University of Brasov, Romania

The notion of generalized Lagrange space should be geometrically considered as a generalized metric space $M^n=(M,g_{ij}(x,y))$. A theory of invariant Finsler spaces was given by M. Matsumoto and R. Miron with important applications. The notion of non-holonomic space was introduced by Gh. Vranceanu in [VR]. The Vranceanu type invariant frames and the invariant geometry of second order Lagrange spaces was studied by the author in [P3]. The purpose of the present paper is to study the invariant geometry for a generalized Lagrange space endowed with a Berwald-Moor metric. We introduce distinct non-holonomic frames on the two components of the Whitney's decomposition. This will determine a non-holonomic coordinates system on the total space $TM$ and thus its geometry can be studied with methods analogous to the mobile frame. We obtain, in this manner, invariant connections, curvatures and torsions, and the fundamental equations in this theory. Also we can construct the invariant frames so that, with respect to them, the metric of the total space can be written in canonical form and in this case we deduce invariant Einstein equations. We mention that the frames introduced here depend on the metric and all the computations are for this metric.

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2005jbl

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Expansion of Complex Number
2005jbk | Y. A. Furman, A. V. Krevetsky  // Mari state technical university, Yoshkar Ola, Russia

The expanded complex numbers are introduced by means of imaginary unit $i$ replacement by one-dimensional on multivariate $3D$ or $7D$ imaginary unit $r$. It is shown, full quaternions and octaves appear as a result of a turn around the material axis $0Re$ plane where $a+ib$ number is set in $4D$ and $8D$ spaces. Rotor-complanar classes of quaternions and the octaves appearing as a result of similar transformations are considered. They represent commutative-associative algebras.

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Thomas precession by pseudoquaternions
2005jbj | D. E. Burlankov, G. B. Malykin  // Nizhny Novgorod State University; Institute of Applied Physics RAS, Nizhny Novgorod, Russia

When a body moves curvilinearly in a plane with a velocity that is comparable to the velocity of light, only three coordinates of the body undergo Lorentz transformation, and the transformation matrix appears to be three-parametric. This enables description of these transformations by pseudoquaternions, Hamilton quaternions slightly modified for pseudo-Euclidean character of the metrics. Their algebraic properties and relation to the Lorentz transformations in a 2+1-dimensional Minkovsky space were determined. We integrated the pseudoquaternion differential equation of continuous transformations at a body's motion along a circular orbit and, as a result, obtained an expression for the value of the Thomas precession.

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Nonclosure of elemenary space-time transformations
2005jbi | Chub V. F.  // Korolev Rocket and Space Corporation "Energia"

The article gives a brief group-theoretical comparative study of three space-time theories: (space-time theory in frames of) classical Newton mechanics, special theory of relativity and author-developed theory based on complex-dual quaternions.

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