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

Let (M, g) be a Riemannian manifold admitting a concurrent-recurrent vector field ν\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\nu$$\end{document}. 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) ν♭\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\nu ^\flat$$\end{document} is invariant along gradient of potential function; (ii) M is Einstein; (iii) the potential vector field is pointwise collinear to concurrent-recurrent vector field ν\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\nu$$\end{document}. 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=-λα{g-ν♭⊗ν♭}\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$Ric=-\frac{\lambda }{\alpha }\{g-\nu ^\flat \otimes \nu ^\flat \}$$\end{document}, where ν♭\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\nu ^\flat$$\end{document} is the canonical 1-form associated to ν\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\nu$$\end{document}.