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Kenmotsu Contact Geometry Through the Lens of $\ast-\boldsymbolκ$-Ricci-Bourguignon Almost Solitons

Lavanya Kumar, Soumendu Roy

TL;DR

The paper introduces the $\ast-\boldsymbol{\kappa}$-Ricci-Bourguignon almost soliton on Kenmotsu manifolds, situating it within the framework of $\ast$-Ricci and Ricci–Bourguignon solitons. It derives the soliton equation $\boldsymbol{\kappa}\mathscr{L}_{\mathcal{V}}h+2\mathbb{T}^*-(2\Omega+\theta\mathcal{R}^*)h=0$ and proves that the star-scalar curvature satisfies $\mathcal{R}^*=\mathcal{R}+4n^2$, which together yield the crucial relation $\Omega=\frac{\theta(\mathcal{R}+4n^2)}{2}$ when $\mathcal{V}$ is the Reeb field. The work further analyzes gradient solitons via a Poisson equation $\Delta(u)= -\frac{(\mathcal{R}+4n^2)}{\boldsymbol{\kappa}}\left[1+\frac{\theta(2n+1)}{2}\right]-\frac{\Omega}{\boldsymbol{\kappa}}(2n+1)$ and identifies special cases $\theta\in\{0,1,2\}$ corresponding to $\ast$-$\boldsymbol{\kappa}$-RBS, $\ast$-RB solitons, and $\ast$-$\boldsymbol{\kappa}$-Einsteinian solitons. The role of torse-forming vector fields is developed, giving explicit $\Omega$-formulas and a classification of torse-generating types that yield $\boldsymbol{\eta}$-Einsteinian structures under suitable conditions. An explicit 5D Kenmotsu example demonstrates the construction and confirms the parameter regimes for contracting/steady/growing solitons, highlighting practical realizations of the theory.

Abstract

This paper focuses on the study of the newly introduced $\ast-\boldsymbolκ$-Ricci-Bourguignon almost soliton pertaining to Kenmotsu structure manifolds. Our analysis concerns the characteristics of this soliton and derive the scalar curvature for a Kenmotsu manifold admitting such a structure. Further, we formulate the corresponding vector fields under the assumption that the manifold supports a $\ast-\boldsymbolκ-$Ricci-Bourguignon soliton. Additionally, we explore applications involving torse-forming vector fields within the framework of the $\ast-\boldsymbolκ-$Ricci-Bourguignon almost soliton on Kenmotsu structure manifolds. To support the theoretical findings, we provide a concrete illustration belonging to a $\ast-\boldsymbolκ-$Ricci-Bourguignon almost soliton in a 5D Kenmotsu structure manifold.

Kenmotsu Contact Geometry Through the Lens of $\ast-\boldsymbolκ$-Ricci-Bourguignon Almost Solitons

TL;DR

The paper introduces the -Ricci-Bourguignon almost soliton on Kenmotsu manifolds, situating it within the framework of -Ricci and Ricci–Bourguignon solitons. It derives the soliton equation and proves that the star-scalar curvature satisfies , which together yield the crucial relation when is the Reeb field. The work further analyzes gradient solitons via a Poisson equation and identifies special cases corresponding to --RBS, -RB solitons, and --Einsteinian solitons. The role of torse-forming vector fields is developed, giving explicit -formulas and a classification of torse-generating types that yield -Einsteinian structures under suitable conditions. An explicit 5D Kenmotsu example demonstrates the construction and confirms the parameter regimes for contracting/steady/growing solitons, highlighting practical realizations of the theory.

Abstract

This paper focuses on the study of the newly introduced -Ricci-Bourguignon almost soliton pertaining to Kenmotsu structure manifolds. Our analysis concerns the characteristics of this soliton and derive the scalar curvature for a Kenmotsu manifold admitting such a structure. Further, we formulate the corresponding vector fields under the assumption that the manifold supports a Ricci-Bourguignon soliton. Additionally, we explore applications involving torse-forming vector fields within the framework of the Ricci-Bourguignon almost soliton on Kenmotsu structure manifolds. To support the theoretical findings, we provide a concrete illustration belonging to a Ricci-Bourguignon almost soliton in a 5D Kenmotsu structure manifold.
Paper Structure (5 sections, 8 theorems, 67 equations)

This paper contains 5 sections, 8 theorems, 67 equations.

Table of Contents

  1. Introduction
  2. Preliminaries on Contact Metric Structures
  3. Kenmotsu Metrics Admitting $\ast - \boldsymbol{\kappa}$-Ricci--Bourguignon Almost Solitons ($\ast - \boldsymbol{\kappa}\mathrm{RBS}$)
  4. On the Role arising from Torse-Generating tangent Fields in $\ast$-$\boldsymbol{\kappa}$-Ricci–Bourguignon Almost Soliton on Kenmotsu-type Manifolds
  5. "Explicit Illustration belonging to a 5D Kenmotsu Structure Satisfying the $\ast$-$\boldsymbol{\kappa}$-Ricci-Bourguignon Almost Soliton Equation"

Key Result

Theorem 3.1

If the given tensor metric $h$ for a $(2n+1)$ odd dim-space Kenmotsu-type Manifold admits $\ast$- $\boldsymbol{\kappa}$-Ricci Bourguignon soliton($\ast-\boldsymbol{\kappa} RBS$) $(h,\zeta, \Omega,\theta)$, in which $\zeta$ equals canonical reeb vector field, it follows that geometric soliton becomes

Theorems & Definitions (10)

  • Definition 1.1
  • Theorem 3.1
  • Corollary 3.2
  • Theorem 3.3
  • Definition 3.4
  • Theorem 3.5
  • Theorem 3.6
  • Theorem 3.7
  • Theorem 4.1
  • Corollary 4.2