In this paper, we initiate the study of impact of the existence of a unit vector \(\nu \), called a concurrent-recurrent vector field, on the geometry of a Riemannian manifold. Some examples of these vector fields are provided on Riemannian manifolds, and basic geometric properties of these vector fields are derived. Next, we characterize Ricci solitons on 3-dimensional Riemannian manifolds and gradient Ricci almost solitons on a Riemannian manifold (of dimension n) admitting a concurrent-recurrent vector field. In particular, it is proved that the Riemannian 3-manifold equipped with a concurrent-recurrent vector field is of constant negative curvature \(-\alpha ^2\) when its metric is a Ricci soliton. Further, it has been shown that a Riemannian manifold admitting a concurrent-recurrent vector field, whose metric is a gradient Ricci almost soliton, is Einstein.

在本文中，我们开始研究单位向量\(\nu \)的存在对黎曼流形的几何形状的影响，称为并发循环向量场。在黎曼流形上提供了这些矢量场的一些示例，并导出了这些矢量场的基本几何特性。接下来，我们表征 3 维黎曼流形上的 Ricci 孤子和允许并发循环向量场的黎曼流形（n维）上的梯度 Ricci 几乎孤子。特别地，证明了配备并发循环向量场的黎曼三流形具有恒定的负曲率\(-\alpha ^2\)当它的度量是 Ricci 孤子时。此外，已经表明，允许并发循环向量场的黎曼流形是爱因斯坦，其度量是梯度 Ricci，几乎是孤子。