arXiv subjects

G. S. Asanov

Publications and source records attributed to G. S. Asanov.

At least 19 recordsLinked to original sources

arXiv2018

Finsleroids with three axes in dimension $N=3$

The Conclusive Theorem has been established to determine the dependence of the three-axes positive-definite Finsleroid metric functions $F$ on the Finsleroid azimuthal angle $θ$ in the three-dimensional case $N=3$, provided that the condition of the angle-separation in the involved characteristic functions is implied. The complete set of algebraic and differential equations is derived in all rigor which are necessary and sufficient in order that the function $F$ belong to the class. It proves possible to solve the equations and obtain the explicit dependence of the involved characteristic functions on the angle $θ$.

math.DG↗

arXiv2017

Two-axes pseudo-Finsleroid metrics: general overview and angle-regular solution

The class of the two-axes pseudo-Finslerian metrics which is specified by the condition of the angle-separation in the involved characteristic functions is proposed and studied. The complete Total Set of algebraic and differential equations is derived in all rigor which are necessary and sufficient in order that a pseudo-Finsleroid metric function belong to the class. It proves possible to solve the equations of the set. The angle-regular solution of the Finsleroid-in-pseudo-Finsleroid type is found and described in detail.

math.DG↗

arXiv2015

Pseudo-Finsleroid metric function of spatially anisotropic relativistic type

The paper contributes to the important and urgent problem to extend the physical theory of space-time in a Finsler-type way under the assumption that the isotropy of space is violated by a single geometrically distinguished spatial direction which destroys the pseudo-Euclidean geometric nature of the relativistic metric and space. It proves possible to retain the fundamental geometrical property that the indicatrix should be of the constant curvature. Similar property appears to hold in the three-dimensional section space. The last property was the characteristic of three-dimensional positive-definite Finsleroid space proposed and developed in the previous work, so that the present paper lifts that space to the four-dimensional relativistic level. The respective pseudo-Finsleroid metric function is indicated. Numerous significant tensorial and geometrical consequences have been elucidated. \ses {\bf Keywords:} Finsler metrics, relativistic spaces.

math.GM↗

arXiv2011

Finsler connection preserving the two-vector angle under the indicatrix-inhomogeneous treatment

The Finsler spaces in which the tangent Riemannian spaces are conformally flat prove to be characterized by the condition that the indicatrix is a space of constant curvature. In such spaces the Finslerian normalized two-vector angle can be explicated from the respective two-vector angle of the associated Riemannian space. Therefore the way is opening to propose explicitly the connection preserving the angle even at the indicatrix-inhomogeneous level, that is, when the indicatrix curvature value ${\mathcal C}_{\text{Ind.}} $ is permitted to be an arbitrary smooth function of the indicatrix position point $x$. The connection obtained is metrical with the deflection part which is proportional to the gradient of the function $H(x)$ entering the equality ${\mathcal C}_{\text{Ind.}} \equiv H^2.$ Also the connection is covariant-constant. When the transitivity of covariant derivative is used, from the commutators of covariant derivatives the associated curvature tensor is found. Various useful representations have been developed. The Finsleroid space has been explicitly outlined.

math.DG↗

arXiv2010

Finsler connection preserving angle in dimensions $N\ge3$

We show that if a Finsler space is conformally automorphic to a Riemannian space and the automorphism is positively homogeneous with respect to tangent vectors, then the indicatrix of the Finsler space is a space of constant curvature. In this case, the Finslerian two-vector angle can explicitly be found, which gives rise to simple and explicit representation for the connection preserving the angle in the indicatrix-homogeneous case. The connection is metrical and the Finsler space is obtainable from the Riemannian space by means of the parallel deformation. Since also the transitivity of covariant derivative holds, in such Finsler spaces the metrical non-linear angle-preserving connection is the respective export of the metrical linear Riemannian connection. From the commutators of covariant derivatives the associated curvature tensor is found. In case of the ${\cal FS}$-space, the explicit example of the conformally automorphic transformation can be developed, which entails the explicit connection coefficients and the metric function of the Finsleroid type.

math.DG↗

arXiv2009

Finsleroid gives rise to the angle-preserving connection

The Finslerian unit ball is called the {\it Finsleroid} if the covering indicatrix is a space of constant curvature. We prove that Finsler spaces with such indicatrices possess the remarkable property that the tangent spaces are conformally flat with the conformal factor of the power dependence on the Finsler metric function. It is amazing but the fact that in such spaces the notion of the two-vector angle defined by the geodesic arc on the indicatrix can readily be induced from the Riemannian space obtained upon the conformal transformation, which opens up the straightforward way to induce also the connection coefficients and the concomitant curvature tensor. Thus, we are successfully inducing the Levi-Civita connection from the Riemannian space into the Finsleroid space, obtaining the isometric connection. The resultant connection coefficients are not symmetric. However, the metricity condition holds fine, that is, the produced covariant derivative of the Finsleroid metric tensor vanishes identically. The particular case underlined by the axial Finsleroid of the ${\mathbf\cF\cF^{PD}_{g}}$-type is explicitly evaluated in detail. Keywords: Finsler metrics, connection, curvature, conformal properties.

math.DG↗

arXiv2009

Finslerian angle-preserving connection in two-dimensional case. Regular realization

We show that the metrical connection can be introduced in the two-dimensional Finsler space such that entailed parallel transports along curves joining points of the underlying manifold keep the two-vector angle as well as the length of the tangent vector, thereby realizing isometries of tangent spaces under the parallel transports. The curvature tensor is found. In case of the Finsleroid-regular space, constructions possess the $C^{\infty}$-regular status globally regarding the dependence on tangent vectors. Many involved and important relations are explicitly derived. Keywords: Finsler metrics, angle, connection, curvature tensors

math.DG↗

arXiv2008

Pseudo-Finsleroid spatial-anisotropic relativistic space

The pseudo-Finsleroid relativistic metric was constructed upon assuming that the involved vector field $b_i$ is time-like. In the present paper it is shown that the metric admits just the alternative counterpart in which the field is space-like. The entailed pseudo-Finsleroid-spatial framework is systematically described. We face on various remarkable properties, including the constant curvature of the associated indicatrix, the explicit Hamiltonian function, transparent presentations for the angle and scalar product. The spray coefficients are found to be of a rather simple structure. The Berwald case is attractively realized. Interesting conformal properties are stemming. {\bf Keywords:} Finsler metrics, relativistic spaces, spray coefficients.

math.DG↗

arXiv2008

Finsleroid-regular space. Landsberg-to-Berwald implication

By performing required evaluations, we show that in the Finsleroid-regular space the Landsberg-space condition just degenerates to the Berwald-space condition (at any dimension number $N\ge2$). Simple and clear expository representations are obtained. Due comparisons with the Finsleroid-Finsler space are indicated. Keywords: Finsler metrics, spray coefficients, curvature tensors.

math.DG↗

arXiv2007

Finsleroid-regular space: curvature tensor, continuation of gravitational Schwarzschild metric

The method of simple straightforward calculation of the curvature tensor of the Finsleroid--regular space is indicated. The Schwarzschild metric which underlines the gravitational field produced by static spherical-symmetric body is shown to be uniquely extended to the Finslerian domain upon a consistent treatment of the pseudo-Finsleroid axis vector field $b_i$ to be the field of the time variable. Keywords: Finsler metrics, gravitational equations, curvature tensors.

math-ph↗

arXiv2007

Finsleroid-regular space developed. Berwald case

The Finsleroid--Finsler space becomes regular when the norm $||b||=c$ of the input 1-form $b$ is taken to be an arbitrary positive scalar $c(x) < 1$. By performing required direct evaluations, the respective spray coefficients have been obtained in a simple and transparent form. The adequate continuation into the regular pseudo-Finsleroid domain has been indicated. The Finsleroid-regular Berwald space is found under the assumptions that the Finsleroid charge is a constant and the 1-form $b$ is parallel. Keywords: Finsler metrics, spray coefficients, curvature tensors.

math.DG↗

arXiv2007

Finsleroid-Finsler Space of Involutive Case

The Finsleroid-Finsler space is constructed over an underlying Riemannian space by the help of a scalar $g(x)$ and an input 1-form $b$ of unit length. Explicit form of the entailed tensors, as well as the respective spray coefficients, is evaluated. The involutive case means the framework in which the characteristic scalar $g(x)$ may vary in the direction assigned by $b$, such that $dg=μb$ with a scalar $μ(x)$. We show by required calculation that the involutive case realizes through the $A$-special relation the picture that instead of the Landsberg condition $\dot A_{ijk}=0$ we have the vanishing $\dot{\al}_{ijk}=0$ with the normalized tensor $\al_{ijk}=A_{ijk}/||A||$. Under the involutive condition, the derivative tensor $A_{i|j}$ and the curvature tensor $R^i{}_k$ have explicitly been found, assuming the input 1-form $b$ be parallel. Key words: Finsler metrics, spray coefficients, curvature tensors.

math.DG↗

arXiv2007

Finsleroid Corrects Pressure and Energy of Universe. Respective Cosmological Equations

The Hubble constant proves to be the pseudo-Finsleroid--Landsberg factor. The covariantly conserved pseudo-Finsleroid--gravitational tensor is explicitly found after evaluating the respective Finsleroid--case curvature tensor and required contractions in attentive way. The equations arisen involve one parameter g of extension which measures the Finslerian deviation of the curvature of the indicatrix of unit vectors. The vector field b^i(x) of the axes of the pseudo--Finsleroids is naturally identified to the field of average velocity vectors of matter of the universe. The consistent (and unique) continuation of the Robertson--Walker metric, and hence the Friedmann metrics, in the Finslerian domain with respect to the parameter g is arisen. The cosmological pressure and energy density prove to be linear functions of g^2, so that the presence of the negative pressure seems to be not necessary to get the agreement with the observed negative nature of deceleration parameter. We clarify the explicit structure of all the involved tensorial objects.

math-ph↗

arXiv2006

Finsleroid--Finsler Parallelism

The Finsleroid--induced scalar product, and hence the angle, proves to remain unchanged under the Finsleroid--type parallel transportation of involved vectors in the Landsberg case. The two--vector extension of the Finsleroid metric tensor is proposed.

math.DG↗

arXiv2006

Finsleroid--Finsler Space and Spray Coefficients

In the previous work, the notion of the Finsleroid--Finsler space have been formulated and the necessary and sufficient conditions for the space to be of the Landsberg type have been found. In the present paper, starting with particular spray coefficients, we demonstrate how the Landsberg condition can explicitly appear in case of the Finsleroid--type metric function. Calculations are supplementing by a convenient special Maple--program. The general form of the associated geodesic spray coefficients is presented for such metric function under the condition of constancy of the Finsleroid charge. Key words: Finsler geometry, metric spaces, spray.

math.DG↗

arXiv2006

Finsleroid--Finsler Space with Berwald and Landsberg Conditions

We formulate the notion of the Finsleroid--Finsler space, including the positive--definite as well as indefinite cases. The associated concepts of angle, scalar product, and the distance function are elucidated. If the Finsleroid--Finsler space is of Landsberg type, then the Finsleroid charge is a constant. The Finsleroid--Finsler space proves to be a Berwald space if and only if the Finsleroid--axis 1-form is parallel with respect to the associated Riemannian metric and, simultaneously, the Finsleroid charge is a constant. The necessary and sufficient conditions for the Finsleroid--Finsler space to be of the Landsberg type are found, which are explicit and simple. The structure of the associated curvature tensors has been elucidated.

math.DG↗

arXiv2005

Finslerian grounds for four--directional anisotropic kinematics

Upon straightforward four--directional extension of the special--relativistic two--dimensional transformations to the four--dimensional case we lead to convenient totally anisotropic kinematic transformations, which prove to reveal many remarkable group and invariance properties. Such a promise is shown to ground the basic manifold with the Finslerian fourth-root metric function to measure length of relativistic four--vectors. Conversion to the framework of relativistic four--momentum is also elucidated. The relativity principle is strictly retained. An interesting particular algebra for subtraction and composition of three-dimensional relative velocities is arisen. The correspondence principle is operative in the sense that at small relative velocities the transformations introduced tend approximately to ordinary Lorentzian precursors. The transport synchronization remains valid. Abbreviation RF will be used for (inertial) reference frames. {\bf Keywords:} special relativity, invariance, Finsler geometry.

math-ph↗

arXiv2005

Finslerian metric function of totally anisotropic type. Relativistic aspects

The work focuses upon the relativistic and geometric properties of the space--time endowed tentatively with the metric function of the Berwald--Moor type. The zero curvature of indicatrix is a remarkable property of the approach. We demonstrate how the associated geodesic equations can be solved in a transparent way, thereby obtaining possibility to introduce unambiguously the distance, angle, and scalar product. We find convenient indicatrix representation for the associated tetrads and, by attributing to them naturally the general meaning of the bases proper of inertial reference frames, elucidate respective fundamental kinematic relations, including the extensions of Lorentz transformations and velocity subtraction and composition laws. The invariance group for the metric tensor is found.

math-ph↗