On non-gradient $$(m,\rho )$$-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 $$(m,\rho )$$ -quasi-Einstein structure on a metric manifold. First, prove that if K-contact or Sasakian manifold $$M^{2n+1}$$ admits closed structure, then it is an Einstein constant scalar curvature $$2n(2n+1)$$ , for particular case—a non-Sasakian $$(k,\mu -contact structure—it locally isometric to product Euclidean space $${\mathbb {R}}^{n+1}$$ sphere $$S^n$$ 4. Next, compact H-contact whose potential vector field V collinear Reeb field, $$\eta $$ -Einstein