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Hemangi Madhusudan Shah

Publications and source records attributed to Hemangi Madhusudan Shah.

At least 19 recordsLinked to original sources
arXiv2026

Non-compact Ricci Solitons of Finite Volume with Potential Field of Constant Length

We study non-compact Ricci solitons of finite volume whose potential vector field has constant length. Under the assumptions that the scalar curvature is constant along the integral curves of the potential field and that a natural divergence term is integrable on the unit tangent bundle, we prove that such Ricci solitons are necessarily trivial. As applications, we characterise $3$-dimensional Ricci solitons of constant length and of finite volume on almost $α$-cosymplectic manifolds. In fact, we show that these spaces are indeed compact quotients of Euclidean spaces. We also study complete, non-compact and finite volume contact metric manifolds in dimension $3$ admitting a Reeb vector field as a Ricci soliton. We show that it is a compact spherical space form. We include some examples to illustrate the necessity of the finite-volume and constant-length assumptions.

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arXiv2026

Ricci solitons from the perspectives of energy function

This article explores to what extent the geometry of gradient Ricci solitons extends to non-gradient Ricci solitons. The primary tool is the energy function $E$ of the soliton. We study consequences of various bounds on $E$. Under mild assumptions on the scalar curvature, we prove a weighted $L^1$-Liouville type theorem for both the usual Laplacian and the drifted Laplacian $Δ_V$ associated to soliton vector field $V$, the former of which implies that Ricci solitons with bounded energy function have at most one nonparabolic end. Finally, we show that the measure $e^{-E} \mathrm{d} \mathrm{Vol}_{g}$ is finite for complete shrinking Ricci solitons, partially generalizing a result of Aaron Naber. As a consequence, non-gradient shrinking Ricci solitons also have finite fundamental groups, as in the gradient case.

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arXiv2026

Mixed Killing Vector Fields on Cigar Ricci-Bourguignon Solitons

In this article, we study mixed Killing vector fields, defined by the condition $L_V L_V g = f\,L_V g$, on Cigar Ricci--Bourguignon solitons. While conformal vector fields are always mixed Killing, the converse fails in flat and open cylinders with base manifold geometries, where the mixed Killing class is infinite-dimensional. We establish a rigidity phenomenon for Cigar Ricci--Bourguignon solitons: any complete steady almost gradient Ricci--Bourguignon soliton on a surface with positive curvature is, up to homothety, Hamilton's Cigar soliton. We then characterise complete mixed Killing fields and show that locally any mixed Killing field is the sum of a rotational Killing field and a mixed Killing radial field. Finally, we establish that the dimension of the vector space of complete mixed Killing fields of Cigar Ricci--Bourguignon solitons is $5$. Moreover, we explicitly determine a basis. Our results show that Cigar Ricci--Bourguignon solitons exhibit behaviour completely different from that of Euclidean space. Finally, we provide a complete description of the geodesic structure of Cigar Ricci--Bourguignon solitons.

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arXiv2026

A Constructive Approach to Infinitesimal Conformal Rigidity on Complex Hyperbolic Space

We prove that every conformal vector field on the complex hyperbolic space $\mathbb{C}H^n$ is Killing for all $n\ge 2$. Although this rigidity is classically known, our proof is entirely different in nature: it is local, analytic, and fully constructive. Our approach is local, analytic, and constructive: we view $\mathbb{C}H^2$ through its solvable Lie group model and express the conformal Killing equation as an explicit system of partial differential equations. By solving this system completely, we show that any conformal vector field must be determined by a Killing field. The analysis in complex dimension $2$ naturally extends to arbitrary $n$, yielding a unified and fully explicit proof of this rigidity phenomenon.

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arXiv2026

Infinitesimal Conformal Rigidity on Damek-Ricci Spaces

We show that every conformal vector field on a Damek-Ricci space is necessarily Killing, establishing a strong form of infinitesimal conformal rigidity. Although this rigidity phenomenon is classically known in the Einstein setting, our proof follows a completely different approach. We formulate the conformal Killing condition as an explicit system of partial differential equations on the solvable Lie group model and analyze it directly. This local and analytic method yields a constructive proof of rigidity without relying on global arguments or transformation groups.

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arXiv2026

Finite Volume Einstein Finsler Warped Product Manifolds of Non-positive or Non-negative Scalar Curvature

The notion of warped product plays an important role in Riemannian geometry moreover in geodesic metric spaces. The warped product was first introduced by Bishop and O'Neill to study Riemannian manifolds of negative curvature.Warped products have been mainly used to construct new examples of Riemannian manifolds with prescribed curvature conditions. This construction can be extended for Finslerian metrics with some minor restrictions. This is motivated by Asanov's papers, where some models of relativity theory are described through the warped product of Finsler metrics. These metrics are in the form of $(α,β)$-metrics, which are the generalization of the Randers metrics; which are being asymmetric Finsler metrics in four-dimensional space-time. The product was later extended to the warped product case of Finsler manifolds by the work of Kozma, Peter and Verge.

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arXiv2025

Characterizations of Almost Ricci Bourguignon Solitons

In this paper, we revisit the study of almost Ricci-Bourguignon solitons by clarifying their position in the broader context of Einstein-type metrics. Motivated by known rigidity results for compact almost Ricci solitons, we aim to identify conditions under which a compact almost RB-soliton is trivial or exhibits special geometric properties. We compare our results with classical theorems of Barros and Ribeiro, and explain explicitly how our work extends or complements these earlier findings.

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arXiv2025

Geometry of Clairaut Riemannian warped product submersions

In this paper, we introduce and study the concept of \textit{Clairaut Riemannian warped product submersions} between Riemannian warped product manifolds. By generalizing the notion of Clairaut Riemannian submersions to the setting of Riemannian warped product submersions, we define such submersions via a warping function satisfying a Clairaut relation along geodesics. We establish necessary and sufficient conditions under which a Riemannian warped product submersion satisfies the Clairaut condition, showing that it holds if and only if the girth function defining the Clairaut condition has a horizontal gradient, one component of the fibers is totally geodesic, and the other is totally umbilical with mean curvature vector governed by the warping function. We examine the geometric consequences of this structure, study the harmonicity conditions, and the behavior of the Weyl tensor, etc. Additionally, we illustrate the theory with several non-trivial examples. In the latter part of the paper, we explore a detailed study of the curvature behavior of such submersions. Explicit formulas for the Riemannian, Ricci, and sectional curvature tensors of the source space are derived in terms of the geometry of the target and fiber manifolds, as well as the warping and girth functions. These computations provide geometric insight into how warping and the Clairaut condition affect curvature properties, such as conformal flatness and the non-positivity of certain mixed curvatures. We also analyze the conditions for a trivial warping of the source manifold and for the fibers to be locally symmetric. Furthermore, the Einstein condition has been explored in various scenarios. Finally, we also extend and answer a question posed in [2] to the setting of Clairaut warped product submersion.

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arXiv2025

Some Characteristics of Almost $ω$-Bach Solitons

In this article, we introduce $ω$-Bach tensor corresponding to one form $ω$ and correspondingly introduce almost $ω$-Bach solitons, thereby generalizing the existing notion of Bach tensor and almost Bach solitons. We characterize almost $ω$-Bach solitons, when the potential vector field of the soliton generates an infinitesimal harmonic transformation or is an affine conformal vector field, or is a projective vector field or is a Killing vector field, when the $ω$-Bach tensor is divergence free, or is a harmonic $1$ form or is a Killing $1$-form. We generalize some of the results obtained by P. T. Ho and A. Ghosh. One of the main results of this paper is that we explicitly find some of the gradient almost $ω$-Bach solitons on the product manifolds ${\mathbb S}^2\times{\mathbb H}^2$, $\mathbb{R}^2\times{\mathbb H}^2$ and $\mathbb{R}^2\times{\mathbb S}^2$. Our gradient almost $ω$-Bach solitons generalize the almost Bach solitons on $\mathbb{R}^2\times{\mathbb H}^2$ and $\mathbb{R}^2\times{\mathbb S}^2$ found by P. T. Ho. Moreover, finding of our gradient almost $ω$-Bach solitons on ${\mathbb S}^2\times{\mathbb H}^2$ is a novel one and complements to the existing almost Bach solitons described by P. T. Ho.

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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 $-\frac{1}{4}$.

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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.

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arXiv2023

Asymptotically harmonic manifolds of dimension 3 with minimal horospheres

In [14], it was shown that, if M is a 3-dimensional asymptotically harmonic with minimal horospheres, then M is flat. However, there is a gap in the proof of this paper. In this paper, we provide the correct proof of the result. Thus we complete the classification of asymptotically harmonic manifolds of dimension 3: An asymptotically harmonic manifold of dimension 3 is either a flat or real hyperbolic space.

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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.

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arXiv2023

Nature of Some Solitons on Almost coKähler Manifolds and Asymptotically Harmonic Manifolds

In this research, we study the nature of $η$-Einstein and gradient $η$-Einstein soliton in the framework of almost coKähler manifolds and $(κ, μ)$-almost coKähler manifolds. We find some expressions for scalar curvature of the almost coKähler manifold admitting $η$-Einstein soliton in various cases. We also prove that if a $(κ, μ)$-almost coKähler manifold admits a gradient $η$-Einstein soliton, then either the manifold is coKähler, or $N(κ)$-almost coKahler, or the soliton is trivial. We present an example which validates our results. Finally, we investigate the asymptotically harmonic manifolds admitting non-trivial Ricci solitons and show that they exhibit rigid behavior if for example, scalar curvature attains maximum.

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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.

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arXiv2023

Some Solitons on Homogeneous Almost $α$-Cosymplectic $3$-Manifolds and Harmonic Manifolds

In this paper, we investigate the nature of Einstein solitons, whether it is steady, shrinking or expanding on almost $α$-cosymplectic $3$-manifolds. We also prove that a simply connected homogeneous almost $α$-cosymplectic $3$-manifold, admitting a contact Einstein soliton, is an unimodular semidirect product Lie group. Finally, we show that a harmonic manifold admits a Ricci soliton if and only if it is flat.

math.GM↗
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.

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