The parallel rank of a submanifold of Euclidean space

Abstract

The aim of this paper is to make some remarks about the theory of parallel immersions in ℝn. We show that the existence of parallels to a curve in ℝndepends on the values of the integrals of then− 1 curvature functions along the curve. We then give examples of immersions ofS1in ℝ3and of immersions of codimension 2 with local parallel rank zero, thereby answering certain questions in [1].