Generalized Ricci solitons on Riemannian manifolds admitting concurrent-recurrent vector field
Tóm tắt
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
$$\nu$$
.
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