arXiv subjects

Salah G. Elgendi

Publications and source records attributed to Salah G. Elgendi.

10 recordsLinked to original sources

arXiv2026

Spherically symmetric Finsler metrics satisfying the $σ$T-condition

In this paper, we present a complete characterization of spherically symmetric Finsler metrics that satisfy the $σT$-condition. We further investigate the subclass of such metrics within the Landsberg category and identify the precise conditions under which spherically symmetric Finsler metrics satisfying the $T$-condition must also be Berwaldian. In addition, we construct new non-regular solutions to the classical unicorn problem, providing explicit examples of Landsberg metrics that are not Berwaldian.

math.DG↗

arXiv2026

The T-tensor of spherically symmetric Finsler metrics

This paper is devoted to the study of the T-tensor associated with a spherically symmetric Finsler metric $F=uϕ(r,s)$ on $\mathbb{R}^n$. We derive a general expression for the T-tensor in terms of the scalar function $ϕ(r, s)$ and its partial derivatives. Furthermore, we characterize all spherically symmetric Finsler metrics satisfying the so-called T-condition, that is, those for which the T-tensor vanishes. In addition, we obtain the formula for the mean Cartan tensor and demonstrate that all spherically symmetric Finsler metrics of dimension $n \geq 3$, with a non-zero mean Cartan tensor are quasi-C-reducible.

math.DG↗

arXiv2024

Parallel one forms on special Finsler manifolds

In this paper, we investigate the existence of parallel 1-forms on specific Finsler manifolds. We demonstrate that Landsberg manifolds admitting a parallel 1-form have a mean Berwald curvature of rank at most $n-2$. As a result, Landsberg surfaces with parallel 1-forms are necessarily Berwaldian. We further establish that the metrizability freedom of the geodesic spray for Landsberg metrics with parallel 1-forms is at least $2$. We figure out that some special Finsler metrics do not admit a parallel 1-form. Specifically, no parallel 1-form is admitted for any Finsler metrics of non-vanishing scalar curvature, among them the projectively flat metrics with non-vanishing scalar curvature. Furthermore, neither the general Berwald's metric nor the non-Riemannian spherically symmetric metrics admit a parallel 1-form. Consequently, we observe that certain $(α,β)$-metrics and generalized $(α,β)$-metrics do not admit parallel 1-forms.

math.DG↗

arXiv2024

The geometry of geodesic invariant functions and applications to Landsberg surfaces

In this paper, for a given spray $S$ on an $n$-dimensional manifold $M$, we investigate the geometry of $S$-invariant functions. For an $S$-invariant function $¶$, we associate a vertical subdistribution $\V_¶$ and find the relation between the holonomy distribution and $\V_¶$ by showing that the vertical part of the holonomy distribution is the intersection of \ok{all spaces $\V_{\F_S}$ associated to $\F_S$ where $\F_S$} is the set of all Finsler functions that have the geodesic spray $S$. As an application, we study the Landsberg Finsler surfaces. We prove that a Landsberg surface with $S$-invariant flag curvature is Riemannian or has a vanishing flag curvature. We show that for Landsberg surfaces with non-vanishing flag curvature, the flag curvature is $S$-invariant if and only if it is constant, in this case, the surface is Riemannian. Finally, for a Berwald surface, we prove that the flag curvature is $H$-invariant if and only if it is constant.

math.DG↗

arXiv2024

Finsler surfaces with vanishing $T$-tensor

In this paper, for Finsler surfaces, we prove that the T-condition and $σT$-condition coincide. For higher dimensions $n\geq 3$, we illustrate by an example that the T-condition and $σT$-condition are not equivalent. We show that the non-homothetic conformal change of a Berwald (resp. a Landsberg) surface is Berwaldian (resp. Landsbergian) if and only if the $σT$-condition is satisfied. By solving the Landsberg's PDE, we classify all Finsler surfaces satisfying the T-condition, or equivalently the $σT$-condition. Some examples are provided and studied.

math.DG↗

arXiv2023

An intrinsic proof of Numata's theorem on Landsberg spaces

In this paper, we study the unicorn's Landsberg problem from an intrinsic point of view. Precisely, we investigate a coordinate-free proof of Numata's theorem on Landsberg spaces of scalar curvature. In other words, following the pullback approach to Finsler geometry, we prove that all Landsberg spaces of dimension $n\geq 3$ of non-zero scalar curvature are Riemannian spaces of constant curvature.

math.DG↗

arXiv2023

A note on "On the classification of Landsberg spherically symmetric Finsler metrics"

In this paper, we prove that all spherically symmetric Landsberg surfaces are Berwaldian. We modify the classification of spherically symmetric Finsler metrics, done by the author in [S. G. Elgendi, On the classification of Landsberg spherically symmetric Finsler metrics, Int. J. Geom. Methods Mod. Phys. 18 (2021)], of Berwald type of dimension $n\geq 3$. Precisely, we show that all Berwald spherically symmetric metrics of dimension $n\geq 3$ are Riemannian or given by a certain formula. As a simple class of Berwaldian metrics, we prove that all spherically symmetric metrics in which the function $ϕ$ is homogeneous of degree $-1$ in $r$ and $s$ are Berwaldian.

math.DG↗

arXiv2022

A new look at Finsler surfaces and the Landsberg's PDE

In this paper, we introduce a new look at Finsler surfaces. Landsberg surfaces are Finsler surfaces that are solutions of a system of non-linear partial differential equations. Considering the unicorn's Landsberg problem, we reduce this system to a single non-linear PDE which we call the Landsberg's PDE. By making use of the new look of Finsler surfaces, we solve the Landsberg's PDE and get a class of solutions. Moreover, we show that these solutions and their conformal transformations are Berwaldain.

math.DG↗

arXiv2019

Metrizability of holonomy invariant projective deformation of sprays

In this paper, we consider projective deformation of the geodesic system of Finsler spaces by holonomy invariant functions: Starting by a Finsler spray $S$ and a holonomy invariant function $P$, we investigate the metrizability property of the projective deformation $\widetilde{S}=S-2λP C$. We prove that for any holonomy invariant nontrivial function $P$ and for almost every value $λ\in R$, such deformation is not Finsler metrizable. We identify the cases where such deformation can lead to a metrizable spray: in these cases, the holonomy invariant function is necessarily one of the principal curvatures of the geodesic structure.

math.DG↗

arXiv2016

Freedom of h(2)-variationality and metrizability of sprays

In this paper we are investigating variational homogeneous second order differential equations by considering the questions of how many different variational principles exist for a given spray. We focus our attention on h(2)-variationality; that is, the regular Lagrange function is homogeneous of degree two in the directional argument. Searching for geometric objects characterizing the degree of freedom of h(2)-variationality of a spray, we show that the holonomy distribution generated by the tangent direction to the parallel translations can be used to calculate it. As a working example, the class of isotropic sprays is considered.

math.CA↗