Three-Dimensional Riemannian Manifolds and Ricci Solitons

We characterize the three-dimensional Riemannian manifolds endowed with a semi-symmetric metric rho-connection if its Riemannian metrics are Ricci and gradient Ricci solitons, respectively. It is proved that if a three-dimensional Riemannian manifold equipped with a semi-symmetric metric rho-connection admits a Ricci soliton, then the manifold possesses the constant sectional curvature -1 and the soliton is expanding with lambda = -2. Next, we study the gradient Ricci solitons in such a manifold. Finally, we construct a non-trivial example of a three-dimensional Riemannian manifold endowed with a semi-symmetric metric rho-connection admitting a Ricci soliton and validate our some results.