Differential geometry of vector fields and its applications

Recently, rectifying submanifolds in Euclidean space have been defined and classified. These studies are actually a generalization of the studies for rectifying curves in 3-dimensional Euclidean space. In this study, firstly, the concept of rectifying submanifold in an arbitrary Riemannian manifold is given. Then, torqued vector fields in Riemannian manifolds are examined in detail and a classification of Riemannian manifolds defined by this vector field is given. Then rectifying submanifolds in the Riemannian manifold given with this vector field are explained. In addition, Ricci solitons with torqued potential vector fields are studied. Finally, some results about Ricci solitons are given and their applications are examined in detail.