Geometry of Ricci solitons admitting a new geometric vector field

In the present paper, we introduce a new geometric vector field (it will be called semi-Killing field) on semi-Riemannaian manifolds. A complete classification of semi-Killing vector fields on 3-dimensional Walker manifolds will be derived. Then, we study Ricci solitons admitting this new vector field (called semi-Killing vector field) as their potential. In Riemannain setting, we prove that Ricci solitons with semi-Killing potential vector field are Einstein. Our results show that such Lorentzian solitons have constant scalar curvature. Finally, application of this new structure in theoretical physics has been investigated.