On submanifolds with a parallel normal vector field in spaces of constant curvature

In this paper, we describe normal vector fields of a special form along geodesic lines on $n$-dimensional submanifolds of $(n+p)$-dimensional spaces of constant curvature, in particular, fields of normal curvature and normal torsion of a submanifold at a point in a given direction. We study submanifolds such that these normal vector fields are parallel in the normal connection along their geodesic lines.