Browsing by Author "Hemangi Madhusudan Shah"

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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 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.
The Funk–Finsler Structure on the Unit Disc in the Hyperbolic Plane
(Birkhauser, 2024) Ashok Kumar; Hemangi Madhusudan Shah; Bankteshwar Tiwari
In this paper, we construct a 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 the Ricci curvature in the Klein unit disc. © The Author(s), under exclusive licence to Springer Nature Switzerland AG 2024.
The isoperimetric problem in Randers plane
(Institute of Mathematics, University of Debrecen, 2024) Arti Sahu Gangopadhyay; Ranadip Gangopadhyay; Hemangi Madhusudan Shah; Bankteshwar Tiwari
In 1947, Busemann observed that a Minkowski circle need not be a solution of the isoperimetric problem in a Minkowski plane. Li and Mo recently showed that the Euclidean circles centred at the origin in a unit ball with the Funk metric are solutions of the isoperimetric problem [9]. In this paper, we construct a class of Randers planes in which any Euclidean circle, centered at the origin in R2, turns out to be a local minimum of the isoperimetric problem with respect to the various well-known volume forms in Finsler geometry. As a consequence, it turns out that the Euclidean circles centred at the origin are solutions of the isoperimetric problem in a Randers type Minkowski plane. © 2024 Institute of Mathematics, University of Debrecen. All rights reserved.
The isoperimetric problem in Randers Poincaré disc
(World Scientific, 2024) Arti Sahu Gangopadhyay; Ranadip Gangopadhyay; Hemangi Madhusudan Shah; Bankteshwar Tiwari
It is known that a simply connected Riemann surface satisfies the isoperimetric equality if and only if it has constant Gaussian curvature. In this paper, we show that the circles centered at origin in the Randers Poincaré disc satisfy the isoperimetric equality with respect to different volume forms however, these Randers metrics do not necessarily have constant (negative) flag curvature. In particular, we show that Osserman's result [12] of the Riemannian case cannot be extended to the Finsler geometry as such. © 2024 World Scientific Publishing Company.
The isoperimetric problem in Randers Poincaré disc
(World Scientific, 2025) Arti Sahu Gangopadhyay; Ranadip Gangopadhyay; Hemangi Madhusudan Shah; Bankteshwar Tiwari
It is known that a simply connected Riemann surface satisfies the isoperimetric equality if and only if it has constant Gaussian curvature. In this paper, we show that the circles centered at origin in the Randers Poincaré disc satisfy the isoperimetric equality with respect to different volume forms however, these Randers metrics do not necessarily have constant (negative) flag curvature. In particular, we show that Osserman's result [12] of the Riemannian case cannot be extended to the Finsler geometry as such. © World Scientific Publishing Company.