Differential Geometry of Rectifying Submanifolds

A space curve in a Euclidean 3-space E 3 is called a rectifying curve if its position vector field always lies in its rectifying plane. This notion of rectifying curves was introduced by the author in Amer. Math. Monthly 110 (2003), no. 2, 147-152. In this present article, we introduce and study the notion of rectifying submanifolds in Euclidean spaces. In particular, we prove that a Euclidean submanifold is rectifying if and only if the tangential component of its position vector field is a concurrent vector field. Moreover, rectifying submanifolds with arbitrary codimension are completely determined.