Generalized Ricci solitons on Riemannian manifolds admitting concurrent-recurrent vector field

Abstract

Let (M, g) be a Riemannian manifold admitting a concurrent-recurrent vector field $\nu$. We prove that if the metric g is a generalized Ricci soliton such that the potential field V is a conformal vector field, then M is Einstein. Next we show that if the metric of M is a gradient generalized Ricci soliton, then either of these three occurs: (i) $\nu ^\flat$ is invariant along gradient of potential function; (ii) M is Einstein; (iii) the potential vector field is pointwise collinear to concurrent-recurrent vector field $\nu$. Finally, we investigate gradient generalized Ricci soliton on a Riemannian manifold (M, g) admitting a unit parallel vector field, and in this case we show that if g is a non-steady gradient generalized Ricci soliton, then the Ricci tensor satisfies $Ric=-\frac{\lambda }{\alpha }\{g-\nu ^\flat \otimes \nu ^\flat \}$, where $\nu ^\flat$ is the canonical 1-form associated to $\nu$.