In this paper, we study Riemann solitons on Sasakian 3-manifolds. We prove that if a Sasakian 3-manifold (M,g) admits a Riemann soliton with a potential vector field V where divV is constant, then g is homothetic to a Berger sphere. It is also shown that any Sasakian 3-manifold (M,g) that admits a Riemann soliton with potential vector field ??, where ? is Reeb vector field and ? is a smooth function on M, is an Einstein manifold. Also, we prove that if a Sasakian 3-manifold (M,g) is is Einstein, Einstein-semisymmetric, projectively flat, or ?-projectively flat manifold then (M,g) satisfies the Riemann soliton equation. Finally, we prove that if a Sasakian 3-manifold (M,g) has a gradient Riemann soliton with potential vector field ?f, then ? must be constant. Additionally, if a Sasakian 3-manifold (M,g) admits a Riemann soliton (M,g,V,?) 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.
Jafari et al. (Wed,) studied this question.