A characterization of ξ-submanifolds in Euclidean space

Submanifolds in R^n Euclidean space constitute one of the important topics of differential geometry. In this area, it includes curves, surfaces, and all hypersurfaces. Submanifolds are defined with the aid of an isometric immersion x. This function is a vector valued function on R^n and is known as the position vector of the submanifold. Various studies have been carried out on the properties of the position vector. When this vector is decomposed into its tangent and normal components, different geometric properties of each component emerge. An important vector of submanifolds is the mean curvature vector. Thanks to this vector, it is possible to characterize submanifolds. If the normal component of the position vector and the mean curvature vector are linner dependent, these submanifolds are called self-shrinking. Such submanifolds are very important for soliton theory in physics. If the sum of these two vector fields is expressed as a xi vector field then the submanifold is expressed as the xi-submanifold. Rotation and toroidal submanifolds have an important place in modern differential geometry. Especially in R^3, rotation surfaces are very important in computer aided geometric design. In 4-dimensional Euclidean space, rotation surfaces form a richclass. In this study, the case of rotation submanifolds in R^n Euclidean space as xi-submanifolds is discussed.