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Dhriti Sundar Patra

Publications and source records attributed to Dhriti Sundar Patra.

11 recordsLinked to original sources
arXiv2026

Characterizations of weak almost ${\mathcal S}$-manifolds with curvature properties

Weak metric structures, introduced by Rovenski and Wolak in 2022, extend Yano's $f$-structure and almost contact metric structure. In this paper, we investigate curvature phenomena of weak almost ${\cal S}$-manifolds (w.a.$\,\cal S$-manifolds) focusing on the $f$-$(κ,μ)$-nullity condition and its special case $R_{X,Y}\,ξ=0$. We establish several results that generalize known rigidity theorems for almost ${\cal S}$-manifolds. First, using the partial Ricci flow, we obtain dynamical characterizations of $\cal S$-manifolds: starting from a w.a.$\,\cal S$-structure satisfying the curvature condition of $\cal S$-manifolds or the $f$-$(1,μ)$-nullity condition, the flow evolves the structure exponentially fast toward an $\cal S$-structure. This extends results of Cappelletti Montano and Di Terlizzi to the weak metric setting. Next, we identify conditions under which a w.a.$\,{\cal S}$-manifold admits a bi-Legendrian structure with totally geodesic foliations. Finally, for w.a.$\,\cal S$-manifolds with $κ=μ=0$, we prove a splitting theorem in which one factor is flat, generalizing classical results for almost $\cal S$-geometry. These findings have consequences for the theory of Sasakian and $\cal S$- manifolds, the geometry of bi-Legendrian structures, and the behavior of weak metric contact manifolds under curvature constraints.

math.DG↗
arXiv2026

On the splitting of weak nearly ${\cal C}$-manifolds

The interest of mathematicians in metric $f$-manifolds, in particular, almost contact metric manifolds, is motivated by the study of the geometry and dynamics of contact foliations, as well as their applications in physics. Weak metric $f$-manifolds, defined by V. Rovenski and R. Wolak (2022), open a new perspective on classical theory of $f$-manifolds and discover new applications. In this paper, we study manifolds of this type, called weak nearly ${\cal C}$-manifolds, which generalize almost ${\cal C}$-manifolds. We find conditions under which a $(2n+s)$-dimensional weak nearly ${\cal C}$-manifold becomes locally a Riemannian product, and characterize $(4+s)$-dimensional weak nearly ${\cal C}$-manifolds. The consequences of these theorems present new results for nearly ${\cal C}$-manifolds.

math.DG↗
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.

math.DG↗
arXiv2022

On the rigidity of the Sasakian structure and characterization of cosymplectic manifolds

We introduce new metric structures on a smooth manifold (called "weak" structures) that generalize the almost contact, Sasakian, cosymplectic, etc. metric structures $(φ,ξ,η,g)$ and allow us to take a fresh look at the classical theory. We demonstrate this statement by generalizing several well-known results. We prove that any Sasakian structure is rigid, i.e., our weak Sasakian structure is homothetically equivalent to a Sasakian structure. We show that a weak almost contact structure with parallel tensor $φ$ is a weak cosymplectic structure and give an example of such a structure on the product of manifolds. We find conditions for a vector field to be a weak contact infinitesimal transformation.

math.DG↗
arXiv2021

Geometry of almost contact metrics as almost $*$-Ricci solitons

In the present paper, we give some characterizations by considering $*$-Ricci soliton as a Kenmotsu metric. We prove that if a Kenmotsu manifold represents an almost $*$-Ricci soliton with the potential vector field $V$ is a Jacobi along the Reeb vector field, then it is a steady $*$-Ricci soliton. Next, we show that a Kenmotsu matric endowed an almost $*$-Ricci soliton is Einstein metric if it is $η$-Einstein or the potential vector field $V$ is collinear to the Reeb vector field or $V$ is an infinitesimal contact transformation.

math.DG↗
arXiv2020

On non-gradient $(m,ρ)$-quasi-Einstein contact metric manifolds

Many authors have studied Ricci solitons and their analogs within the framework of (almost) contact geometry. In this article, we thoroughly study the $(m,ρ)$-quasi-Einstein structure on a contact metric manifold. First, we prove that if a $K$-contact or Sasakian manifold $M^{2n+1}$ admits a closed $(m,ρ)$-quasi-Einstein structure, then it is an Einstein manifold of constant scalar curvature $2n(2n+1)$, and for the particular case -- a non-Sasakian $(k,μ)$-contact structure -- it is locally isometric to the product of a Euclidean space $\RR^{n+1}$ and a sphere $S^n$ of constant curvature $4$. Next, we prove that if a compact contact or $H$-contact metric manifold admits an $(m,ρ)$-quasi-Einstein structure, whose potential vector field $V$ is collinear to the Reeb vector field, then it is a $K$-contact $η$-Einstein manifold.

math.DG↗
arXiv2020

Almost $η$-Ricci solitons on Kenmotsu manifolds

In this paper we characterize the Einstein metrics in such broader classes of metrics as almost $η$-Ricci solitons and $η$-Ricci solitons on Kenmotsu manifolds, and generalize some results of other authors. First, we prove that a Kenmotsu metric as an $η$-Ricci soliton is Einstein metric if either it is $η$-Einstein or the potential vector field $V$ is an infinitesimal contact transformation or $V$ is collinear to the Reeb vector field. Further, we prove that if a Kenmotsu manifold admits a gradient almost $η$-Ricci soliton with a Reeb vector field leaving the scalar curvature invariant, then it is an Einstein manifold. Finally, we present new examples of $η$-Ricci solitons and gradient $η$-Ricci solitons, which illustrate our results.

math.DG↗
arXiv2018

The Critical Point Equation on Kenmotsu and almost Kenmotsu manifolds

In this paper, we have studied the critical point equation (shortly, CPE) within the frame-work of Kenmotsu and almost Kenmotsu manifold satisfying certain nullity conditions. First, we prove that a complete Kenmotsu metric satisfies the CPE is Einstein and locally isometric to the hyperbolic space H2n+1. In case of Kenmotsu manifolds, it is possible to determine the potential function explicitly (locally). We also provide some examples of Kenmotsu and almost Kenmotsu manifolds that satisfies the CPE.

math.DG↗
arXiv2018

The k-almost Ricci solitons and contact geometry

The aim of this article is to study the k-almost Ricci soliton and k-almost gradient Ricci soliton on contact metric manifold. First, we prove that if a compact K-contact metric is a k-almost gradient Ricci soliton then it is isometric to a unit sphere S2n+1. Next, we extend this result on a compact k-almost Ricci soliton when the flow vector field X is contact. Finally, we study some special types of k-almost Ricci soliton where the potential vector field X is point wise collinear with the Reeb vector field ξ of the contact metric structure.

math.DG↗
arXiv2017

The Critical Point Equation And Contact Geometry

In this paper, we consider the CPE conjecture in the frame-work of $K$-contact and $(κ, μ)$-contact manifolds. First, we prove that if a complete $K$-contact metric satisfies the CPE is Einstein and is isometric to a unit sphere $S^{2n+1}$. Next, we prove that if a non-Sasakian $ (κ, μ) $-contact metric satisfies the CPE, then $ M^{3} $ is flat and for $ n > 1 $, $ M^{2n+1} $ is locally isometric to $ E^{n+1}\times S^{n}(4)$.

math.DG↗