On the classification of quadratic harmonic morphisms between Euclidean spaces

Abstract

We give a classification of quadratic harmonic morphisms between Euclidean spaces (Theorem 2.4) after proving a Rank Lemma. We also find a correspondence between umbilical (Definition 2.7) quadratic harmonic morphisms and Clifford systems. In the case ${\mathbb{R}}^4⟶{\mathbb{R}}^3$, we determine all quadratic harmonic morphisms and show that, up to a constant factor, they are all bi-equivalent (Definition 3.2) to the well-known Hopf construction map and induce harmonic morphisms bi-equivalent to the Hopf fibration ${\mathbb{S}}^3⟶{\mathbb{S}}^2$.

13 pages, AMS-Latex