arXiv subjects

Mehdi Rafie-Rad

Publications and source records attributed to Mehdi Rafie-Rad.

4 recordsLinked to original sources

arXiv2023

On projective symmetries on Finsler Spaces

There are two definitions of Einstein-Finsler spaces introduced by Akbar-Zadeh, which we will show is equal along the integral curves of $I$-invariant projective vector fields. The sub-algebra of the $C$-projective vector fields, leaving the $H$-curvature invariant, has been studied extensively. Here we show on a closed Finsler space with negative definite Ricci curvature reduces to that of Killing vector fields. Moreover, if an Einstein-Finsler space admits such a projective vector field then the flag curvature is constant. Finally, a classification of compact isotropic mean Landsberg manifolds admitting certain projective vector fields is obtained with respect to the sign of Ricci curvature.

math.DG↗

arXiv2019

On the C-projective vector fields on Randers spaces

A characterization of the C-projective vector fields on a Randers spaces is presented in terms of a recently introduced non-Riemannian quantity defined by Z. Shen and denoted by ${\bfΞ}$; It is proved that the quantity ${\bfΞ}$ is invariant for C-projective vector fields. Therefore, the dimension of the algebra of the C-projective vector fields on an $n$-dimensional Randers space is at most $n(n+2)$. The generalized Funk metrics on the $n$-dimensional Euclidean unit ball $\mathbb{B}^n(1)$ are shown to be explicit examples of the Randers metrics with a C-projective algebra of maximum dimension $n(n+2)$. Then, it is also proved that an $n$-dimensional Randers space has a C-projective algebra of maximum dimension $n(n+2)$ if and only if it is locally Minkowskian or (up to re-scaling) locally isometric to the generalized Funk metric. A new projective invariant is also introduced.

math.DG↗

arXiv2012

Brachistochrone Pursuit - Evasion Games and Riemann-Finsler Geometry

A geometric approach to differential game theory is illustrated. The parallel pursuit is considered as a two-player zero-sum differential game. The optimal strategies of each player is designed based on Riemann-Finsler geometry. Our approach incorporates a closed loop optimal control and the presentation is familiar with a Brachistochrone type problem.

math.OC↗

arXiv2011

On the Projective Algebra of Randers Metrics of Constant Flag Curvature

The collection of all projective vector fields on a Finsler space $(M, F)$ is a finite-dimensional Lie algebra with respect to the usual Lie bracket, called the projective algebra denoted by $p(M,F)$ and is the Lie algebra of the projective group $P(M,F)$. The projective algebra $p(M,F=α+β)$ of a Randers space is characterized as a certain Lie subalgebra of the projective algebra $p(M,α)$. Certain subgroups of the projective group $P(M,F)$ and their invariants are studied. The projective algebra of Randers metrics of constant flag curvature is studied and it is proved that the dimension of the projective algebra of Randers metrics constant flag curvature on a compact $n$-manifold either equals $n(n+2)$ or at most is $\frac{n(n+1)}{2}$.

math.DG↗