Isometry classification of cubic homogeneous 3-dimensional forms

The problem of classification of cubic homogeneous Finslerian 3D metrics with respect to their isometries is considered. It is shown, that there are 6 different general affine types of such metrics. Algebras of isometries are presented in apparent kind together with their affine-invariant properties. Interrelation between symmetries and projective classifyings is discussed.Comment: Talk presented at the conference Symmetry-2009 (Hungary

Plate-Universe in Multidimensional Elasticity Theory

A number of boundary problems in multidimensional elasticity theory are solved. The solutions can be treated as the simplest cosmological models. Some specific properties of the solutions and experimental consequences of the theory are discussed.Comment: 20 pages, 20 figure

Classical long-range interacting N-particle configurations and its applications

We consider classical N-particle system with arbitrary central pair potential. Mechanical equilibrium condition in spherically-symmetric case leads to a nonlinear integro-differential equation for concentration n(r). For special state equation $p=an^{\gamma}(1+bn)$, original integro-differential equation transforms into purely integral one. Its solution (under b=0) is written through a row over interaction parameter. Physical conditions for its convergency is discussed. For power-like potential kernal of integral operator is calculated in apparent kind. The cases of Coulomb and harmonic potentials are considered separately up to a third order. General scheme of application of the theory for some astrophysical and cosmological problems is presented. Model system with spherical potential pits is considered. By perturbation theory first virial correction $\delta n$ is calculated.Comment: Latex-2e, 14 pages, talk presented at school-seminar "Theoretical problems of gravity and cosmology", Russia, Ulyanovsk, september 2000. Shortened variant to be published in "Gravitation and Cosmology" 200

5-Dimensional Covariance and Generation of Solutions of Einstein Equations

A generation procedure, based on the 5-dimensional covariance of the Kaluza-Klein theory, is developed. The procedure allows one to obtain exact solutions of the 4-dimensional Einstein equations with electromagnetic and scalar fields from vacuum 5-dimensional solutions using special 5-dimensional coordinate transformations. Relations between the physical properties of the resulting solutions and invariant geometrical properties of the generating Killing vectors are found out.Comment: Latex 2.09, 6 page

Multidimensional generalization of Kasner solution

Full generalization of Kasner metric for the case of $n+1$ dimensions and $m\le n+1$ essential variables is obtained. Any solution is defined by the corresponding constant matrix of Kasner parameters. This parameters form in euclidian space Casner hyperspheres and are connected by additional conditions. General properties of obtained solutions are analyzed.Comment: 9 pages, LaTe

Deformational Structures on Smooth Manifolds

Deformational structures, in many aspects generalizing standard elasticity theory, are investigated in abstract form. Within free deformational structures we define algebra of deformations, classify them by its special properties, define motions and conformal motions together with deformational decomposition of manifolds, generalizing isometry of Riemannian spaces and consider some physical examples. In frame of dynamical deformational structures we formulate variational procedure for evolutional and static cases together with boundary conditions, derive dynamical (equilibrium in static case) equations, consider perturbative approach and perform deformational realization of the well known classical field-theoretical topics: strings and branes theories, classical mechanics of solids, gravity and Maxwell electrodynamics.Comment: This is developed variant of talk, given at the 5-th ICGA conference (Russia,Moscow, October 2001). To be published in Acta Phys. Polonica 2002-2003; Latex2e, 31 pages, 4 figure

Clasical solids dynamics as 4D statics of elastic strings

Variational principle for a solid in classical mechanics is formulated in terms of a thin elastic 4D bar strain in Minkowsky events space of special relativity. It is shown, that the sum of elastic 4-energies of weak twist and bending under some identifications takes the form of classical non-relativistic action for a solids dynamics. The necessary conditions on 4D bar parameters and elastic constants, providing validity of Newton mechanics, are found.Comment: Latex 2e, 23 pages, 4 figures, talk at Ulyanovsk school-seminar 2000. Accepted for printing in Nuovo Cimento

Space-time as multidimensional elastic plate

It is suggested, that a curved 4-dimensional space-time manifold is a strained elastic plate in multidimensional embedding space-time. Its thicknesses along extradimensions are much less than 4-dimensional sizes. Reduced 4-dimensional free energy density of the strained plate in a weak strain case is similar to GR Lagrangian density of a gravitational field for the particular value of the Poisson coefficient of the plate. Dynamical equations of the theory are obtained by variation of the multidimensional free energy over displacement vector components. In general case they are inhomogeneous bewave equations.Comment: 12 pages, 3 TexCad figures, Latex2

Are different geometries really that different?

Here is presented a concept of centrogeometry which can be seen as a combination of the concept of point-like observer with an idea of Poincar\'{e}'s that different geometries are principally equivalent. As it is to be shown later, all centrogeometries are obtained from each other by general deformation (i.e. active coordinate transformations). Isometries of centrogeometries are equivalent to those of the Euclidean centrogeometry as described by common diffeomorphisms of the Euclidean spheres. There are discussed physical aspects of centrogeometry in the context of chronogeometry, mechanics and cosmology.Comment: 13 pages, 2 figures, talk that will be presented at PIRT-2009, July (Russia), one misprint is removed, formulation of the theorem is made more clea

Geometrization of perfect fluid in 5-D Kaluza-Klein theory

General formulation of geometrization matter problem by scalar field $\phi =\sqrt{-G_{55}}$ with the help of possibilities of classical 5-D Kaluza-Klein theory is given. Mathematical integrability conditions for such geometrization for the case of perfect fluid are derived.Comment: Latex 2.09, 9 page