Riemann solitons on Sasakian 3-manifolds

Abstract

‎In this paper‎, ‎we study the Riemann solitons on Sasakian 3-manifolds‎. ‎We prove that‎, ‎if a Sasakian 3-manifold $(M,g)$ admits‎Riemann soliton with potential vector field $V$ where ${\rm div}V$ is constant then $g$ is homothetic to a Berger sphere‎. ‎Then‎, it is proven that any Sasakian 3-manifold $(M,g)$ that admits a Riemann soliton with potential vector field $\beta \xi$ where $\xi$‎ ‎is Reeb vector field and $\beta$ is a smooth function on $M$‎, ‎is an Einstein manifold‎. Also‎, ‎we prove that if a Sasakian 3-manifold $(M,g)$ is an Einstein or an Einstein-semisymmetric or projectively flat or‎ ‎$\varphi$-projectively flat manifold then $(M,g)$ satisfies in a Riemann soliton‎. ‎In final‎, ‎we prove that if a Sasakian‎ 3-manifold $(M,g)$ has a gradient Riemann soliton with potential vector field $\nabla f$‎, ‎then $f$ must be constant‎. Additionally‎, ‎if a Sasakian 3-manifold $(M,g)$ admits a RS $(M,g,V,\mu)$ such that $V$ is an infinitesimal contact‎ ‎transformation‎, ‎then the transverse geometry of $M$ is Fano and $V$ is a harmonic infinitesimal automorphism of the contact metric structure‎.