Some characterizations of -Einstein solitons on Sasakian manifolds

Dhriti Sundar Patra

doi:10.4153/s0008439522000078

Abstract

AbstractThe $\rho $ -Einstein soliton is a self-similar solution of the Ricci–Bourguignon flow, which includes or relates to some famous geometric solitons, for example, the Ricci soliton and the Yamabe soliton, and so on. This paper deals with the study of $\rho $ -Einstein solitons on Sasakian manifolds. First, we prove that if a Sasakian manifold M admits a nontrivial $\rho $ -Einstein soliton $(M,g,V,\lambda )$ , then M is $\mathcal {D}$ -homothetically fixed null $\eta $ -Einstein and the soliton vector field V is Jacobi field along trajectories of the Reeb vector field $\xi $ , nonstrict infinitesimal contact transformation and leaves $\varphi $ invariant. Next, we find two sufficient conditions for a compact $\rho $ -Einstein almost soliton to be trivial (Einstein) under the assumption that the soliton vector field is an infinitesimal contact transformation or is parallel to the Reeb vector field $\xi $ .

Highlights

In recent years the pioneering works of R
Bourguignon [4] introduced a perturbed version of the Ricci flow on an n-dimensional Riemannian manifold (M, g), which satisfies the following evolution equation [4] considered a geometric flow of the following type:
Theorem 1.4 If a K-contact manifold (M2n+1, φ, ξ, η, g) admits a ρ-Einstein almost soliton (M, g, V, λ), whose potential vector field V is parallel to the Reeb vector field ξ, V is a Killing vector field, g is trivial (Einstein) and of constant scalar curvature 2n(2n + 1)

Summary

Introduction

In recent years the pioneering works of R. Theorem 1.3 If a compact K-contact manifold (M2n+1, φ, ξ, η, g) admits a ρ-Einstein almost soliton (M, g, V, λ) with the potential vector field V is an infinitesimal contact transformation, V is an infinitesimal automorphism and g is trivial (Einstein) Sasakian and of constant scalar curvature 2n(2n + 1). Theorem 1.4 If a K-contact manifold (M2n+1, φ, ξ, η, g) admits a ρ-Einstein almost soliton (M, g, V, λ), whose potential vector field V is parallel to the Reeb vector field ξ, V is a Killing vector field, g is trivial (Einstein) and of constant scalar curvature 2n(2n + 1). Corollary 1.5 If a compact K-contact manifold (M2n+1, φ, ξ, η, g) admits a ρ-Einstein almost soliton (M, g, V, λ), whose potential vector field V is parallel to the Reeb vector field ξ, V is a Killing vector field, g is trivial (Einstein) Sasakian and of constant scalar curvature 2n(2n + 1). From their result (see Theorem 3.2 in [21]) it follows that if a complete K-contact manifold admits a gradient ρ-Einstein almost soliton, it is compact Einstein Sasakian and isometric to the unit sphere S2n+1

Background

Proof of Main Results