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Ebtsam H. Taha

Publications and source records attributed to Ebtsam H. Taha.

At least 19 recordsLinked to original sources
arXiv2026

On the generalized $m$-Kropina metrics

Generalized $m$-Kropina metrics appear naturally as a spacetime geometry compatible with Lorentz symmetry breaking, leading to useful applications in modified gravity and cosmology. We prove that a generalized $m$-Kropina metric $F$ is an almost rational Finsler metric. Thereby, we study the rationality of its Finslerian geometric objects in the directional variable $y$. For example, its geodesic spray coefficients are rational in $y$. Consequently, we prove that if $F$ is an Einstein metric with $m \notin \mathbb{Z}$, then it is Ricci-flat. Moreover, for $m \in 2 \mathbb{Z}$, the arithmetic nature of $m$ imposes strong rigidity constraints: if $F$ has isotropic mean Berwald curvature, or has relatively isotropic Landsberg curvature, or has almost vanishing $\mathbf{H}$-curvature, then $F$ is weakly Berwaldian, or $F$ is Landsbergian, or $\mathbf{H}=0$, respectively. Furthermore, we show that if $F$ has almost isotropic flag curvature ($m \in 2 \mathbb{Z}$ and $n \geq 3$), then the flag curvature is constant. We, hence, deduce under what conditions a generalized $m$-Kropina metric $F$ becomes an exact solution to "Pfeifer and Wohlfarth's vacuum field equation". Finally, we provide several four-dimensional examples arising in modified gravity and cosmology.

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arXiv2026

Golden Finsler Geometry: Local Properties and Global Deformations

We introduce the concept of a golden Finsler structure on a finite-dimensional smooth manifold $M$ and investigate it from both local (coordinate-based) and global (coordinate-free) perspectives. Locally, we explicitly compute the fundamental metric tensor, establish the positive definiteness condition, and derive the geodesic spray coefficients. Furthermore, we investigate the projective flatness of the golden $(α, β)$-metric and prove the non-existence of almost rational golden $(α, β)$-metrics. Globally, we define the golden Finsler change $\widetilde{F}$ of a base Finsler metric $F$ and examine its geometric properties utilizing a special concurrent $π$-vector field. We explicitly determine how fundamental non-linear structures, including the Barthel and Berwald connections, transform under this change. Finally, we prove that $\widetilde{F}$ and $F$ cannot be projectively related.

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arXiv2025

A Concurrent Generalized Kropina Change

This paper investigates a generalized Kropina metric featuring a specific $π$-form. Start with a Finsler manifold $(M,F)$ admits a concurrent $π$-vector field $\overlineφ$, then, examine the $ϕ$-concurrent generalized Kropina change defined by $\widehat{F}=\frac{F^{m+1}}{Φ^{m}},\,\, Φ^{m}>0$, where $Φ$ represents the corresponding $1$-form. We investigate the fundamental geometric objects associated with $\widehat{F}$ in an intrinsic manner after adopting this modification and present an example of a Finsler metric that admits a concurrent vector field along with $\widehat{F}$. Also, we prove that the geodesic sprays of $F$ and $\widehat{F}$ can never be projectively related. Moreover, we show $\overlineφ$ is not concurrent with respect to $\widehat{F}$. Eventhough, we give a sufficient condition for $\overlineφ$ to be concurrent with respect to $\widehat{F}$. Finally, we prove that the $ϕ$-concurrent generalized Kropina change ($F \longrightarrow \widehat{F}$) preserves the almost rational property of the initial Finsler metric ${F}$.

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arXiv2025

On Generalized Matsumoto Metrics with a Special $π$-form

We explore a generalization of Matsumoto metric intrinsically. Given a Finsler manifold $(M,F)$ which admits a concurrent $π$-vector field $\overlineφ$, we consider the change $\widehat{F}(x,y)=\frac {F^2 (x,y)} {F(x,y)-Φ(x,y)}$, where $Φ$ is the associated concurrent $π$-form with $F(x,y) > Φ(x,y)$ for all $(x,y) \in \T M$. We find the condition under which the generalized $ϕ$-Matsumoto metric $\widehat{F}$ is a Finsler metric. Moreover, the relations between the associated Finslerian geometric objects of $\widehat{F}$ and $F$ are obtained, namely, the relations between angular metric tensors, metric tensors, Cartan torsions, geodesic sprays, Barthel connections (along with its curvature) and Berwald connections. Further, we prove that the Finsler metrics $F$ and $\widehat{F}$ can never be projectively related. Also, a condition for the $π$-vector field $\overlineφ$ to be concurrent with respect to $\widehat{F}$ is acquired. Moreover, an example of a rational Finsler metric admitting a concurrent $π$-vector field together with the associated change $\widehat{F}$ is provided. Finally, we find the conditions that preserve the almost rationality property of a Finsler metric $F$ under the $ϕ$-Matsumoto change.

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arXiv2025

Special anisotropic conformal changes of conic pseudo-Finsler surfaces

This study presents many special anisotropic conformal changes of a conic pseudo-Finsler surface $(M,F)$, such as $C$-anisotropic and horizontal $C$-anisotropic conformal transformations, which reduce to $C$-conformal when the conformal factor is solely position-dependent. Furthermore, we present vertical $C$-anisotropic conformal changes and demonstrate that they are characterized by the property of $(M,F)$ being Riemannian. Additionally, we examine the anisotropic conformal transformation that fulfils the $ϕT$-condition, the horizontal $ϕT$-condition, and the vertical $ϕT$-condition. The first two conditions reduce to the $\boldsymbolσ T$-condition when the conformal factor relies solely on a positional variable. We demonstrate that, under the vertical $ϕT$-condition change, every Landsberg surface is Berwaldian. Thus, the vertical $ϕT$-condition is equivalent to the $T$-condition. Furthermore, we examine the scenario when the anisotropic conformal factor becomes the main scalar of the non-Riemannian surface $(M,F)$. We present an example of a Finslerian Schwarzschild-de Sitter solution having Finslerian spherical symmetry and apply our results to it.

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arXiv2025

Anisotropic conformal change of conic pseudo-Finsler surfaces, II

This paper is a continuation of our investigation of the anisotropic conformal change of a conic pseudo-Finsler surface $(M,F)$, namely, the change $\overline{F}(x,y)=e^{ϕ(x,y)}F(x,y)$ \cite{first paper}. We obtain the relationship between some important geometric objects of $F $ and their corresponding objects of $\overline{F}$, such as Berwald, Landsberg and Douglas tensors, as well as the T-tensor. In contrast to isotropic conformal transformation, under an anisotropic conformal transformation, we find out the necessary and sufficient conditions for a Riemannian surface to be anisotropically conformal transformed to Berwald or Landsberg or Douglas surfaces. Consequently, we determine under what condition the geodesic spray of a two-dimensional pseudo-Berwald metric $\overline{F}$ is Riemann metrizable by a two-dimensional pseudo-Riemannian metric $F$. We show an example of a conformal transformation of a Riemannian metric $F$ that is not geodesically equivalent to a Riemannian metric but is instead Berwaldian. Also, we determine the necessary and sufficient conditions for $F $ to be anisotropically conformally flat (i.e., $\overline{F}$ is Minkowskian). Moreover, we identify the required conditions for preserving the $T$-condition under an anisotropic conformal change. Finally, we establish the necessary conditions for a Riemannian metric to be anisotropically conformal to a Douglas metric.

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arXiv2024

Anisotropic conformal change of conic pseudo-Finsler surfaces, I

The present work is devoted to investigate anisotropic conformal transformation of conic pseudo-Finsler surfaces $(M,F)$, that is, $ F(x,y)\longmapsto \overline{F}(x,y)=e^{ϕ(x,y)}F(x,y)$, where the function $ϕ(x,y)$ depends on both position $x$ and direction $y$, contrary to the ordinary (isotropic) conformal transformation which depends on position only. If $F$ is a pseudo-Finsler metric, the above transformation does not yield necessarily a pseudo-Finsler metric. Consequently, we find out necessary and sufficient condition for a (conic) pseudo-Finsler surface $(M,F)$ to be transformed to a (conic) pseudo-Finsler surface $(M,\overline{F})$ under the transformation $\overline{F}=e^{ϕ(x,y)}F$. In general dimension, it is extremely difficult to find the anisotropic conformal change of the inverse metric tensor in a tensorial form. However, by using the modified Berwald frame on a Finsler surface, we obtain the change of the components of the inverse metric tensor in a tensorial form. This progress enables us to study the transformation of the Finslerian geometric objects and the geometric properties associated with the transformed Finsler function $\overline{F}$. In contrast to isotropic conformal transformation, we have a non-homothetic conformal factor $ϕ(x,y)$ that preserves the geodesic spray. Also, we find out some invariant geometric objects under the anisotropic conformal change. Furthermore, we investigate a sufficient condition for $\overline{F}$ to be dually flat or/and projectively flat. Finally, we study some special cases of the conformal factor $ϕ(x,y)$. Various examples are provided whenever the situation needs.

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arXiv2022

Harmonic Finsler manifolds of $(α, β)$-type

In this paper we construct a new class of harmonic and asymptotically harmonic Finsler manifolds of $(α,β)$-type. This class is defined by a Riemannian metric $α$ and a special 1-form $β$.

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arXiv2021

On Harmonic and Asymptotically Harmonic Finsler Manifolds

In the present paper, we introduce and investigate various types of harmonic Finsler manifolds and find out the interrelation between them. We give some characterizations of such spaces in terms of the mean curvature of geodesic spheres and the Laplacian of the distance function induced by the Finsler structure. We investigate some properties of the Finsler mean curvature of geodesic spheres of different radii. In addition, we prove that certain harmonic Finsler manifolds are of Einstein type and provide a technique to construct harmonic Finsler manifolds of Randers type. Moreover, we give some examples of non-Riemmanian Finsler harmonic manifolds of constant flag curvature and constant $S$-curvature.

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arXiv2021

Tripathi Connection in Finsler Geometry

Adopting the pullback formalism, a new linear connection in Finsler geometry has been introduced and investigated. Such connection unifies all formerly known Finsler connections and some other connections not introduced so far. Also, our connection is a Finslerian version of the Tripathi connection introduced in Riemannian geometry. The existence and uniqueness of such connection is proved intrinsically. An explicit intrinsic expression relating this connection to Cartan connection is obtained. Some generalized Finsler connections are constructed from Tripathi Finsler connection, by applying the P1-process and C-process introduced by Matsumoto. Finally, under certain conditions, many special Finsler connections are given.

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arXiv2021

On Finsler surfaces with certain flag curvatures

In the present paper, we find out necessary and sufficient conditions for a Finsler surface $(M,F)$ to be Landsbregian in terms of the Berwald curvature $2$-forms. We study Finsler surfaces which satisfy some flag curvature $K$ conditions, viz., $V(K)=0,\,\,V(K)= -\mathcal{I}/F^2$ and $V(K)=-\mathcal{I}\,K,$ where $\mathcal{I}$ is the Cartan scalar. In order to do so, we investigate some geometric objects associated with the global Berwald distribution $\mathcal{D}:= \operatorname{span}\{S, H, V:=JH\}$ of a $2$-dimensional Finsler metrizable nonflat spray $S$. We obtain some classifications of such surfaces and show that under what hypothesis these surfaces turn to be Riemannian. The existence of a first integral for the geodesic flow in each case has some remarkable consequences concerning rigidity results. We prove that a Finsler surface with $V(K)= -\mathcal{I}/F^2$ and either $S(K)=0$ or $S(\mathcal{J})=0$ is Riemannian. Further, a Finsler surface with $V(K)=-\mathcal{I}\,K$ and $S(K)=0$ is Riemannian.

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arXiv2021

On almost rational Finsler metrics

We study a special class of Finsler metrics which we refer to as Almost Rational Finsler metrics (shortly, AR-Finsler metrics). We give necessary and sufficient conditions for an AR-Finsler manifold $(M,F)$ to be Riemannian. The rationality of the associated geometric objects such as Cartan torsion, geodesic spray, Landsberg curvature, $S$-curvature, etc is investigated. We prove for a particular subset of AR-Finsler metrics that if $F$ has isotropic $S$-curvature, then its $S$-curvature identically vanishes. Further, if $F$ has isotropic mean Landsberg curvature, then it is weakly Landsberg. Also, if $F$ is an Einstein metric, then it is Ricci-flat. Moreover, we show that Randers metric can not be AR-Finsler metric. Finally, we provide some examples of AR-Finsler metrics and introduce a new Finsler metric which is called an extended $m$-th root metric. We show under what conditions an extended $m$-th root metric is AR-Finsler metric and study its generalized Kropina change.

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arXiv2019

Alternative Lagrangians obtained by scalar deformations

We study non-conservative like SODEs admitting explicit Lagrangian descriptions. Such systems are equivalent to the system of Lagrange equations of some Lagrangian $L$, including a covariant force field which represents non-conservative forces. We find necessary and sufficient conditions for the existence of a differentiable function $Φ:\mathbb{R}\rightarrow\mathbb{R}$ such that the initial system is equivalent to the system of Euler-Lagrange equations of the deformed Lagrangian $Φ(L)$. We give various examples of such deformations.

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arXiv2018

Semi Concurrent vector fields in Finsler geometry

In the present paper, we introduce and investigate the notion of a semi concurrent vector field on a Finsler manifold. We show that some special Finsler manifolds admitting such vector fields turn out to be Riemannian. We prove that Tachibana's characterization of Finsler manifolds admitting a concurrent vector field leads to Riemannain metrics. We give an answer to the question raised in \cite{DWF}: "Is any n-dimensional Finsler manifold $(M,F)$, admitting a non-constant smooth function $f$ on $M$ such that $\frac{\partial f}{\partial x^i}\frac{\partial g^{ij}}{\partial y^k}=0$, a Riemannian manifold?". Various examples for conic Finsler and Riemannian spaces that admit semi-concurrent vector field are presented. Finally, we conjectured that there is no regular Finsler non-Riemannian metric that admits a semi-concurrent vector field. In other words, a Finsler metric admitting a semi-concurrent vector field is necessarily either Riemannian or conic Finslerian.

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arXiv2016

Frobenius integrability and Finsler metrizability for $2$-dimensional sprays

For a $2$-dimensional non-flat spray we associate a Berwald frame and a $3$-dimensional distribution that we call the Berwald distribution. The Frobenius integrability of the Berwald distribution characterises the Finsler metrizability of the given spray. In the integrable case, the sought after Finsler function is provided by a closed, homogeneous $1$-form from the annihilator of the Berwald distribution. We discuss both the degenerate and non-degenerate cases using the fact that the regularity of the Finsler function is encoded into a regularity condition of a $2$-form, canonically associated to the given spray. The integrability of the Berwald distribution and the regularity of the $2$-form have simple and useful expressions in terms of the Berwald frame.

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arXiv2016

Connections in sub-Riemannian geometry of parallelizable distributions

The notion of a parallelizable distribution has been introduced and investigated. A non-integrable parallelizable distribution carries a natural sub-Riemannian structure. The geometry of this structure has been studied from the bi-viewpoint of absolute parallelism geometry and sub-Riemannian geometry. Two remarkable linear connections have been constructed on a sub-Riemannian parallelizable distribution, namely, the Weitzenböck connection and the sub-Riemannian connection. The obtained results have been applied to two concrete examples: the spheres $S^3$ and $S^7$.

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arXiv2016

New Conformal Invariants in Absolute Parallelism Geometry

The aim of the present paper is to investigate conformal changes in absolute parallelism geometry. We find out some new conformal invariants in terms of the Weitzenböck connection and the Levi-Civita connection of an absolute parallelism space.

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