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

TL;DR

This paper investigates the properties of $(m, ho)$-quasi-Einstein structures on contact metric manifolds, revealing conditions under which these manifolds are Einstein or locally isometric to specific product spaces, with implications for their geometric classification.

Contribution

It provides new results characterizing $(m, ho)$-quasi-Einstein contact metric manifolds, especially in the $K$-contact and Sasakian cases, and describes their geometric structures under certain conditions.

Findings

$K$-contact or Sasakian manifolds with closed $(m, ho)$-quasi-Einstein structures are Einstein with constant scalar curvature.

Non-Sasakian $(k,)$-contact structures are locally isometric to a product of Euclidean space and a sphere.

Compact contact manifolds with a potential vector field collinear to the Reeb vector are $K$-contact $ta$-Einstein manifolds.

Abstract

Many authors have studied Ricci solitons and their analogs within the framework of (almost) contact geometry. In this article, we thoroughly study the $(m,\rho)$-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 $(m,\rho)$-quasi-Einstein structure, then it is an Einstein manifold of constant scalar curvature $2n(2n+1)$, and for the particular case -- a non-Sasakian $(k,\mu)$-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 $(m,\rho)$-quasi-Einstein structure, whose potential vector field $V$ is collinear to the Reeb vector field, then it is a $K$-contact $\eta$-Einstein manifold.