On the classification of quadratic harmonic morphisms between Euclidean spaces

TL;DR

This paper classifies quadratic harmonic morphisms between Euclidean spaces, establishes a connection with Clifford systems, and characterizes all such morphisms from 4- to 3-dimensional spaces as related to the Hopf construction.

Contribution

It provides a complete classification of quadratic harmonic morphisms and links umbilical cases to Clifford systems, revealing their relation to the Hopf fibration.

Findings

All quadratic harmonic morphisms from R^4 to R^3 are bi-equivalent to the Hopf map.

Established a Rank Lemma crucial for classification.

Connected umbilical quadratic harmonic morphisms to Clifford systems.

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 $ {\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 ${\Bbb S}^{3} \longrightarrow {\Bbb S}^{2}$.