Differential geometry of rectifying submanifolds

TL;DR

This paper extends the concept of rectifying curves in Euclidean 3-space to rectifying submanifolds in higher-dimensional Euclidean spaces, characterizing them via the tangential component of their position vector fields.

Contribution

It introduces the notion of rectifying submanifolds, proves their characterization through concurrent vector fields, and provides a complete classification for arbitrary codimension.

Findings

Rectifying submanifolds are characterized by their tangential position vector component.

A Euclidean submanifold is rectifying iff its tangential component is a concurrent vector field.

Rectifying submanifolds with any codimension are fully classified.

Abstract

A space curve in a Euclidean 3-space $\mathbb E^3$ 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 {\bf 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.