Quadratic harmonic morphisms and O-systems

TL;DR

This paper introduces O-systems of orthogonal transformations, establishing correspondences with Clifford systems and orthogonal multiplications, and solves existence problems for quadratic harmonic morphisms using these frameworks.

Contribution

It defines O-systems and links them to Clifford systems and orthogonal multiplications, providing a unified approach to existence problems for quadratic harmonic morphisms.

Findings

Established 1-1 correspondences between O-systems, Clifford systems, and orthogonal multiplications.

Solved the existence problem for quadratic harmonic morphisms using the Splitting Lemma.

Analyzed properties of quadratic harmonic morphisms for fixed domain and range spaces.

Abstract

We introduce O-systems (Definition \ref{DO}) of orthogonal transformations of ${\Bbb R}^{m}$, and establish $1-1$ correspondences both between equivalence classes of Clifford systems and that of O-systems, and between O-systems and orthogonal multiplications of the form $\mu :{\Bbb R}^{n} \times {\Bbb R}^{m} \longrightarrow {\Bbb R}^{m} $, which allow us to solve the existence problems both for O-systems and for umbilical quadratic harmonic morphisms (Theorems \ref{ES} and \ref{EU}) simultaneously. The existence problem for general quadratic harmonic morphisms is then solved (Theorem \ref{EG}) by the Splitting Lemma (Lemma \ref{Split}). We also study properties (see, e.g., Theorems \ref{single} and \ref{TL}) possessed by all quadratic harmonic morphisms for fixed pairs of domain and range spaces (\S5).