Abstract
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.
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Naik, D.M. Ricci solitons on Riemannian manifolds admitting certain vector field. Ricerche mat 73, 531–546 (2024). https://doi.org/10.1007/s11587-021-00622-z
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DOI: https://doi.org/10.1007/s11587-021-00622-z