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.

摘要

如果其黎曼度量分别是 Ricci 和梯度 Ricci 孤子，则我们表征具有半对称度量ρ连接的三维黎曼流形。证明了如果配备半对称度量ρ连接的三维黎曼流形允许一个 Ricci 孤子，则该流形具有恒定截面曲率 -1 并且孤子以λ = - 2 扩展。接下来，我们研究这样一个流形中的梯度 Ricci 孤子。最后，我们构造了一个具有半对称度量ρ连接的三维黎曼流形的非平凡示例，该连接承认 Ricci 孤子，并验证了我们的一些结果。