On the classification of quadratic harmonic morphisms between Euclidean spaces

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 $ {Bbb R}^{4}longrightarrow {Bbb 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 ${Bbb S}^{3} longrightarrow {Bbb S}^{2}$.