On collinearity, parallelism and sphericity for pairs of curves

Abstract

The relationship between the notions of collinearity and equichordality [1] is similar to the one between the notions of parallelism and self-parallelism [2]. In [1] some results concerning self-parallelism, equichordality and sphericity were proved.It is therefore natural to look for analogous results but now relating the ideas of parallelism, collinearity and sphericity.This is what we aim at in section 3 of this short note.For simplicity we shall consider only embeddings of $S¹$ into $Rⁿ$ but the proofs work equally well if we replace $S¹$ by a compact, connected, smooth manifold.In section 4 we deal with collinear equichordal embeddings and make a few simple considerations on lengths and chordal areas.