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$\ast$-conformal Einstein solitons on N(k)-contact metric manifolds

Jhantu Das, Kalyan Halder, Soumendu Roy, Arindam Bhattacharyya

Abstract

The main goal of this paper is devoted to N(k)-contact metric manifolds admitting $\ast$-conformal Einstein soliton and also $\ast$-conformal gradient Einstein soliton. In this settings the nature of the manifold, and the potential vector field, potential function of solitons are characterized, and conditions for the $\ast$-conformal Einstein soliton to be expanding, steady, or shrinking are also given. Furthermore, the nature of the potential vector field is evolved when the metric g of N(k)-contact metric manifold satisfies $\ast$-conformal gradient Einstein soliton. Finally, an illustrative example of a N(k)-contact metric manifold is discussed to verify our findings.

$\ast$-conformal Einstein solitons on N(k)-contact metric manifolds

Abstract

The main goal of this paper is devoted to N(k)-contact metric manifolds admitting -conformal Einstein soliton and also -conformal gradient Einstein soliton. In this settings the nature of the manifold, and the potential vector field, potential function of solitons are characterized, and conditions for the -conformal Einstein soliton to be expanding, steady, or shrinking are also given. Furthermore, the nature of the potential vector field is evolved when the metric g of N(k)-contact metric manifold satisfies -conformal gradient Einstein soliton. Finally, an illustrative example of a N(k)-contact metric manifold is discussed to verify our findings.
Paper Structure (5 sections, 6 theorems, 84 equations)

This paper contains 5 sections, 6 theorems, 84 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. $\mathcal{N}(k)$-contact metric manifold admitting $\ast$-conformal Einstein soliton
  4. $\mathcal{N}(k)$-contact metric manifold admitting $\ast$-conformal Einstein gradient soliton
  5. Example

Key Result

Lemma 3.1

(2) A $(2m+1)$-dimensional contact metric manifold $\mathcal{M}$ satisfying $\mathcal{R}(\mathcal{X},\mathcal{Y})\zeta=0$ for any smooth vector fields $\mathcal{X},\mathcal{Y}$ on $\mathcal{M}$ is locally isometric to the Riemannian product of a flat manifold of dimension $(m+1)$ and a manifold of d

Theorems & Definitions (15)

  • Definition 1.1
  • Definition 2.1
  • Definition 2.2
  • Lemma 3.1
  • Lemma 3.2
  • Lemma 3.3
  • proof
  • Theorem 3.1
  • proof
  • Remark 3.2
  • ...and 5 more