Three-dimensional Riemannian manifolds and Ricci solitons

Abstract

We characterize the three-dimensional Riemannian manifolds endowed with a semi-symmetric metric ρ-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 ρ-connection admits a Ricci soliton, then the manifold possesses the constant sectional curvature −1 and the soliton is expanding with λ = −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 ρ-connection admitting a Ricci soliton and validate our some results.