Browsing by Author "Bankteshwar Tiwari"

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A class of Finsler spaces with general (α, β)-metrics
(World Scientific Publishing Co. Pte Ltd, 2019) Bankteshwar Tiwari; Ranadip Gangopadhyay; Ghanashyam Kr. Prajapati
In the present paper, we prove that a general (α,β)-metric F is of isotropic S-curvature if and only if it is of isotropic E-curvature under an extra condition on α and β. © 2019 World Scientific Publishing Company.
A comparative overview of Riemannian and Finsler geometry
(American Mathematical Society, 2025) Bankteshwar Tiwari
The aim of this article is to present a comparative overview of Riemannian and Finsler geometry, starting from some historical developments to the various directions of current research. This includes the discussion on classification of Finsler spaces of constant flag curvature and constant Ricci curvature, cut locus, conjugate locus, Comparison theorems, Gauss-Bonnet-Chern Theorem and sphere theorem in Finsler geometry. The topological, differential and metric structures on Riemannian manifolds in the presence of convex functions have been active fields of research in the second half of the last century. We discuss some of these results on Riemannian manifolds with convex functions and their recently extended analogues on Finsler manifolds. © 2025 American Mathematical Society.
A Note on Best Constants for Weighted Integral Hardy Inequalities on Homogeneous Groups
(Birkhauser, 2024) Michael Ruzhansky; Anjali Shriwastawa; Bankteshwar Tiwari
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≤ q< ∞ . We also calculate the precise value of sharp constants in respective inequalities, improving the result of Ruzhansky and Verma (Proc R Soc A 475:20180310, 2019) in the case of homogeneous groups. © 2023, The Author(s), under exclusive licence to Springer Nature Switzerland AG.
Anisotropic weighted Levin-Cochran-Lee type inequalities on homogeneous Lie groups
(Finnish Mathematical Society, 2025) Michael V. Ruzhansky; Anjali Shriwastawa; Bankteshwar Tiwari
In this paper, we first prove the weighted Levin-Cochran-Lee type inequalities on homogeneous Lie groups for arbitrary weights, quasi-norms, and Lp- and Lq-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. © 2025 The Finnish Mathematical Society.
Hardy inequalities on metric measure spaces, IV: The case p = 1
(Walter de Gruyter GmbH, 2024) Michael Ruzhansky; Anjali Shriwastawa; Bankteshwar Tiwari
In this paper, we investigate the two-weight Hardy inequalities on metric measure space possessing polar decompositions for the case p = 1 and 1 ≤ q < ∞. This result complements the Hardy inequalities obtained in [M. Ruzhansky and D. Verma, Hardy inequalities on metric measure spaces, Proc. Roy. Soc. A. 475 (2019), no. 2223, Article ID 20180310] in the case 1 < p ≤ q < ∞. 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 ≤ q < ∞. © 2024 Walter de Gruyter GmbH, Berlin/Boston.
Isometric Models of the Funk Disc and the Busemann Function
(Birkhauser, 2024) Ashok Kumar; Hemangi Madhusudan Shah; Bankteshwar Tiwari
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 tacitly depends on the measure, in contrast to the Riemannian case. © 2024, The Author(s), under exclusive licence to Springer Nature Switzerland AG.
On Almost Rational Finsler Metrics
(Springer, 2023) Ebtsam H. Taha; Bankteshwar Tiwari
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 some Finsler geometric objects such as Cartan torsion, geodesic spray, Landsberg curvature and S-curvature is investigated. For a particular subfamily of AR-Finsler metrics we have proved that if F has isotropic S-curvature, then the S-curvature vanishes identically; if F has isotropic mean Landsberg curvature, then it is weakly Landsberg; if F is an Einstein metric, then it is Ricci-flat. Moreover, there exists no Randers AR-Finsler metric. Finally, we provide some nontrivial examples of AR-Finsler metrics. © 2023, The Author(s) under exclusive licence to Iranian Mathematical Society.
ON C3-LIKE FINSLER METRICS UNDER RICCI FLOW
(Transilvania University of Brasov 1, 2022) Ranadip Gangopadhyay; Bankteshwar Tiwari
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. © 2022, Transilvania University of Brasov 1. All rights reserved.
On conformal transformation of M-th root finsler metric
(Drustvo Matematicara Srbije, 2020) Bankteshwar Tiwari; Manoj Kumar
The purpose of the present paper is to study the conformal transformation of m-th root Finsler metric. The spray coefficients, Riemann curvature and Ricci curvature of conformally transformed m-th root metrics are shown to be certain rational functions of direction. Further, under certain conditions it is shown that a conformally transformed m-th root metric is locally dually flat if and only if the transformation is a homothety. Moreover the conditions for the transformed metrics to be Einstein and isotropic mean Berwald curvature are also found. © 2020, Drustvo Matematicara Srbije. All rights reserved.
On finsler space with a special (α, β)-metric
(Indian Mathematical Society, 2015) Bankteshwar Tiwari; Manoj Kumar
In the present paper a special (α, β)-metric, which is considered as a generalization of the Rander's metric as well as of the Z. Shen's square metric, has been studied and the conditions for a Finsler space with this special metric to be a Berwald space, a Douglas space and Weakly-Berwald space respectively, have also been found. © Indian Mathematical Society, 2015.
On Finsler warped product metrics with vanishing E-curvature
(Amirkabir University of Technology, 2021) Ranadip Gangopadhyay; Anjali Shriwastawa; Bankteshwar Tiwari
In this paper, we study Finsler warped product metric recently in-troduced by P. Marcal and Z. Shen and find characteristics differential equations for this metric to have vanishing E-curvature. We also prove that if this warped product Finsler metric is projectively flat, then it becomes a Riemannian metric. © 2021, Amirkabir University of Technology. All rights reserved.
On general (α,β)-metrics with some curvature properties
(Tusi Mathematical Research Group (TMRG), 2019) Bankteshwar Tiwari; Ranadip Gangopadhyay; Ghanashyam Kr. Prajapati
In this paper, we study a class of Finsler metric called general (α,β) metrics and obtain an equation that characterizes these Finsler metrics of almost vanishing H-curvature. As a consequence of this result, we prove that a general (α,β)-metric has almost vanishing H-curvature if and only if it has almost vanishing Ξ-curvature. © 2019 Khayyam Journal of Mathematics.
On Generalized Kropina Change of Generalized m-th Root Finsler Metrics
(Springer, 2021) Bankteshwar Tiwari; Manoj Kumar; Akbar Tayebi
In this paper, we study the generalized Kropina change of generalized m-th root metrics. We find the conditions under which the generalized Kropina change of generalized m-th root metrics to be locally projectively flat and locally dually flat. Further, we find the conditions under which such transformed metric of constant flag curvature reduces to a flat Finsler metric. © 2020, The National Academy of Sciences, India.
On generalized Kropina change of m th root Finsler metric
(World Scientific Publishing Co. Pte Ltd, 2017) Bankteshwar Tiwari; Ghanashyam Kr. Prajapati
In the present paper, we consider generalized Kropina change of mth root Finsler metric and prove that it is locally projectively flat if and only if it is locally Minkowskian. We also establish a necessary and sufficient condition under which the generalized Kropina change of mth root metric is locally dually flat. Further it is proved that a generalized Kropina change of mth root metric cannot be conformal to an mth root Finsler metric. © 2017 World Scientific Publishing Company.
On generalized Kropina change of mth root Finsler metrics with special curvature properties
(TUBITAK, 2017) Bankteshwar Tiwari; Ghanashyam Kr. Prajapati
In the present paper, we consider generalized Kropina change of mth root Finsler metrics and prove that every generalized Kropina change of mth root Finsler metrics with isotropic Berwald curvature, isotropic mean Berwald curvature, relatively isotropic Landsberg curvature, and relatively isotropic mean Landsberg curvature reduces to the Berwald metric, weakly Berwald metric, Landsberg metric, and weakly Landsberg metric, respectively. We also show that every generalized Kropina change of mth root Finsler metrics with almost vanishing H-curvature has vanishing H-curvature. © TÜBI˙TAK.
On hypersurfaces of a special semi-C-reducible finsler space
(Indian Mathematical Society, 2014) Bankteshwar Tiwari
The purpose of the present paper is to study hypersurfaces of a Special semi-C-reducible Finsler space. It is found that, if the torsion vector of enveloping space Fn is tangential to its hypersurface Hn-1 and n > 3, then the hypersurface is also a Special semi-C-reducible Finsler space. Finally surfaces in a three dimensional Special semi-C-reducible Finsler space have been studied. The notations and terminologies are referred to the monograph [8]. © Indian Mathematical Society, 2014.
On Kropina change of m-th root Finsler metrics with special curvature properties
(Sciendo, 2018) Bankteshwar Tiwari; Ghanashyam Kr. Prajapati; Ranadip Gangopadhyay
In the present paper, we consider Kropina change of m-th root Finsler metrics and prove that every Kropina change of m-th root Finsler metrics with isotropic Berwald curvature, isotropic mean Berwald curvature, isotropic Landsberg curvature, isotropic mean Landsberg curvature reduces to Berwald metric, weakly Berwald metric, Landsberg metric and weakly Landsberg metric respectively. Also we show that every Kropina change of m-th root Finsler metrics with almost vanishing H-curvature has vanishing H-curvature. © 2018, Sciendo. All rights reserved.
On Minimal Surfaces Immersed in Three Dimensional Kropina Minkowski Space
(Birkhauser, 2022) Ranadip Gangopadhyay; Ashok Kumar; Bankteshwar Tiwari
In this paper we consider a three dimensional Kropina space and obtain a partial differential equation that characterizes 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. © 2021, The Author(s), under exclusive licence to Springer Nature Switzerland AG.
On Minimal Surfaces of Rotations Immersed in Deformed Hyperbolic Kropina Space
(Birkhauser, 2022) Ranadip Gangopadhyay; Ashok Kumar; Hemangi Madhusudan Shah; Bankteshwar Tiwari
In this paper we consider three dimensional upper half space H3 equipped with various Kropina metrics obtained by deformation of hyperbolic metric of H3 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 x3 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. © 2022, The Author(s), under exclusive licence to Springer Nature Switzerland AG.
On normalized semi parallel T'-vector field in finsler space
(2012) Bankteshwar Tiwari
The semi-parallel vector field in Riemannian geometry has been introduced by Fulton [3], whereas in Finsler geometry by Singh and Prasad [9], for instance, concurrent vector fields and concircular vector fields are semi parallel. The purpose of the present paper is to introduce Normalized Semi Parallel T'-vector field in Finsler space and to study the properties of some special Finsler spaces with this vector field. For instance, there in no such vector field in non-Riemannian C-reducible Finsler space. The notations and terminologies are referred to the monograph [5]. © 2012 Academic Publications, Ltd.