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Bankteshwar Tiwari

Publications and source records attributed to Bankteshwar Tiwari.

At least 19 recordsLinked to original sources
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.

math.DG↗
arXiv2026

Finsler structure of the Apollonian weak metric on the unit disc

In this paper, we {\it find} the Finsler structure of the Apollonian weak metric on the open unit disc in $\mathbb{R}^2$, which turns out to be a Randers type Finsler structure and we call it as Apollonian weak-Finsler structure. In fact the Apollonian weak-Finsler structure is the deformation of the hyperbolic Poincaré metric in the unit disc by a closed $1$-form. As a cosequence, the trajectories of the geodesic of this Apollonian weak-Finsler structure pointwise agrees with the geodesic of hyperbolic Poincaré metric in the open unit disc. Further, we explicitly calculate its $S$-curvature, Riemann curvature, Ricci curvature and flag curvature. It turns out that the $S$-curvature of the Apollonian weak-Finsler structure in the unit disc is bounded below by $\frac{3}{2}$, while its flag curvature $K$ satisfies $-\infty< K<-1$, in particular, it becomes a Hadamard manifold.

math.DG↗
arXiv2026

On Locally Dually and Projectively Flat Almost Rational Finsler Metrics

In this paper, we investigate Almost Rational Finsler (AR-Finsler) metrics, a class of Finsler metrics whose fundamental tensor admits a decomposition $g_{ij}(x,y)=η(x,y)a_{ij}(x,y),$ where $η$ is a positive smooth function and $a_{ij}$ is rational in the fiber variables. We derive necessary and sufficient conditions for local dual flatness and local projective flatness of AR-Finsler metrics. Furthermore, we obtain a compatibility relation characterizing AR-Finsler metrics that are simultaneously locally dually flat and locally projectively flat. As an application, we establish a rigidity result for AR-Finsler. Motivated by these results and several known rigidity phenomena in special classes of Finsler metrics, we formulate a conjecture concerning the local Minkowskianity of AR-Finsler metrics that are both locally dually flat and locally projectively flat. Several examples are presented to illustrate the theory.

math.DG↗
arXiv2025

The Funk-Finsler Structure in the Constant Curvature Spaces

In this paper, we {\it find} the infinitesimal structure of Funk-Finsler metric in spaces of constant curvature. We investigate the geometry of this Funk-Finsler metric by explicitly computing its $S$-curvature, Riemann curvature, Ricci curvature, and flag curvature. Moreover, we show that the $S$-curvature of the Funk-Finsler metric in hyperbolic space is bounded above by $\frac{3}{2}$, in spherical space bounded below by $\frac{3}{2}$, and in Euclidean case it is identically equal to $\frac{3}{2}$. Further, we show that the flag curvature of the Funk-Finsler metric in hyperbolic space is bounded above by $-\frac{1}{4}$, in spherical space bounded below by $-\frac{1}{4}$, and in Euclidean case it is identically equal to $-\frac{1}{4}$.

math.DG↗
arXiv2024

On Minkowskian Product Einstein-Finsler spaces

In this paper we study the Minkowskian product Finsler manifolds. More precisely, we prove that if the Minkowskian product Finsler manifold is Einstein then either the product manifold is Ricci flat or both the quotient manifolds are Einstein with same scalar functions.

math.DG↗
arXiv2024

The Funk-Finsler structure on the unit disc in the hyperbolic plane

In this paper, we construct the Funk-Finsler structure in various models of the hyperbolic plane. In particular, in the unit disc of the Klein model, it turns out to be a Randers metric, which is a non-Berwald Douglas metric. Further, using Finsler isometries we obtain the Funk-Finsler structures in other models of the hyperbolic plane. Finally, we also investigate the geometry of this Funk-Finsler metric by explicitly computing the S-curvature, Riemann curvature, flag curvature, and Ricci curvature in the Klein unit disc.

math.DG↗
arXiv2023

Isometric models of the Funk disc and the Busemann function

In this article, we find three isometric models of the Funk disc: Finsler upper half of the hyperboloid of two sheets model, the Finsler band model and the Finsler upper hemi sphere model; and we also find two new models of the Finsler-Poincaré disc. We explicitly describe the geodesics in each model. Moreover, we compute the Busemann function and consequently describe the horocycles in the Funk and the Hilbert disc. Finally, we prove the asymptotic harmonicity of the Funk disc. We also show that, the concept of asymptotic harmonicity of the Finsler manifolds {\it tacitly} depends on the measure, in {\it contrast} to the Riemannian case.

math.DG↗
arXiv2023

Anisotropic weighted Levin-Cochran-Lee type inequalities on homogeneous Lie groups

In this paper, we first prove the weighted Levin-Cochran-Lee type inequalities on homogeneous Lie groups for arbitrary weights, quasi-norms, and $L^p$-and $L^q$-norms. Then, we derive a sharp weighted inequality involving specific weights given in the form of quasi-balls in homogeneous Lie groups. Finally, we also calculate the sharp constants for the aforementioned inequalities.

math.CA↗
arXiv2023

The Isoperimetric Problem in Randers Poincaré Disc

It is known that a simply connected Riemann surface satisfies the isoperimetric equality if and only if it has constant Gaussian curvature. In this article, we show that Randers Poincaré disc satisfies the isoperimetric equality with respect to different volume forms. These metrics do not necessarily have constant (negative) flag curvature. Thus we show that the Osserman's result \cite{RO} in the Riemannian case can not be extended to the corresponding Finslerian case.

math.DG↗
arXiv2022

Hardy inequalities on metric measure spaces, IV: The case $p=1$

In this paper, we investigate the two-weight Hardy inequalities on metric measure space possessing polar decompositions for the case $p=1$ and $1 \leq q <\infty.$ This result complements the Hardy inequalities obtained in \cite{RV} in the case $1< p\le q<\infty.$ The case $p=1$ requires a different argument and does not follow as the limit of known inequalities for $p>1.$ As a byproduct, we also obtain the best constant in the established inequality. We give examples obtaining new weighted Hardy inequalities on homogeneous Lie groups, on hyperbolic spaces and on Cartan-Hadamard manifolds for the case $p=1$ and $1\le q<\infty.$

math.CA↗
arXiv2022

The Isoperimetric Problem In Randers Planes

In this paper, the isoperimetric problem in Randers planes, $(\mathbb{R}^2,F=α+β)$, which are slight deformation of the Euclidean plane $(\mathbb{R}^2,α)$ by suitable one forms $β$, have been studied. We prove that the circles centred at the origin achieves the local maximum area of the isoperimetric problem with respect to well known volume forms in Finsler geometry.

math.DG↗
arXiv2022

On Minimal Surfaces of Revolutions Immersed in Deformed Hyperbolic Kropina Space

In this paper we consider three dimensional upper half space $\mathbb{H}^3 $ equipped with various Kropina metrics obtained by deformation of hyperbolic metric of $\mathbb{H}^3$ through $1$-forms and obtain a partial differential equation that characterizes minimal surfaces immersed in it. We prove that such minimal surfaces can only be obtained when the hyperbolic metric is deformed along $x^3$ direction. Then we classify such minimal surfaces and show that flag curvature of these surfaces is always non-positive. We also obtain the geodesics of this surface. In particular, it follows that such surfaces neither have forward conjugate points nor they are forward complete.

math.DG↗
arXiv2022

A note on best constants for Weighted Integral Hardy inequalities on homogeneous groups

The main aim of this note is to prove sharp weighted integral Hardy inequality and conjugate integral Hardy inequality on homogeneous Lie groups with any quasi-norm for the range $1<p\leq q<\infty.$ We also calculate the precise value of sharp constants in respective inequalities, improving the result of $[19]$ in the case of homogeneous groups.

math.AP↗
arXiv2021

On C3-Like Finsler Metrics Under Ricci Flow

In this paper we have studied the class of Finsler metrics, called C3-like metrics which satisfy the un-normal and normal Ricci flow equation and proved that such metrics are Einstein.

math.DG↗
arXiv2021

On minimal surfaces immersed in three dimensional Kropina Minkowski space

In this paper we consider a three dimensional Kropina space and obtain the partial differential equation that characterizes a minimal surfaces with the induced metric. Using this characterization equation we study various immersions of minimal surfaces. In particular, we obtain the partial differential equation that characterizes the minimal translation surfaces and show that the plane is the only such surface.

math.DG↗
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.

math.DG↗
arXiv2020

Minimal surfaces in three-dimensional Matsumoto space

In this paper we consider the Matsumoto metric $F=\frac{α^2}{α-β}$, on the three dimensional real vector space and obtain the partial differential equations that characterize the minimal surfaces which are graphs of smooth functions and then we prove that plane is the only such surface. We also obtain the partial differential equation that characterizes the minimal translation surfaces and show that again plane is the only such surface.

math.DG↗