A note on Almost Riemann Soliton and gradient almost Riemann soliton

Abstract

The quest of the offering article is to investigate almost Riemann soliton and gradient almost Riemann soliton in a non-cosymplectic normal almost contact metric manifold M 3 . Before all else, it is proved that if the metric of M 3 is Riemann soliton with divergence-free potential vector field Z, then the manifold is quasi-Sasakian and is of constant sectional curvature -λ, provided α, β = constant. Other than this, it is shown that if the metric of M 3 is ARS and Z is pointwise collinear with ξ and has constant divergence, then Z is a constant multiple of ξ and the ARS reduces to a Riemann soliton, provided α, β =constant. Additionally, it is established that if M 3 with α, β = constant admits a gradient ARS (γ, ξ, λ), then the manifold is either quasi-Sasakian or is of constant sectional curvature −(α 2 − β 2 ). At long last, we develop an example of M 3 conceding a Riemann soliton.