Quadratic harmonic morphisms and O-systems

Ou, Ye-Lin

Annales de l'Institut Fourier, Tome 47 (1997) no. 2, pp. 687-713.

Nous présentons les O-systèmes (Définition 3.1) des transformations orthogonales de ${ℝ}^m$ et nous établissons des correspondances à la fois entre les classes d’équivalence des systèmes de Clifford et celles des O-systèmes et les multiplications orthogonales de la forme $\mu :{ℝ}^n\times {ℝ}^m\to {ℝ}^m$, ce qui nous permet de résoudre les problèmes d’existence simultanément pour les O-systèmes et pour les morphismes harmoniques quadratiques ombilicaux. Le problème d’existence pour les morphismes quadratiques harmoniques généraux est alors résolu par le “Splitting Lemma” . Nous avons également étudié les propriétés possédées par tous les morphismes harmoniques quadratiques pour les paires fixes d’espaces de domaines et co-domaines.

We introduce O-systems (Definition 3.1) of orthogonal transformations of ${ℝ}^m$, and establish $ℝ$ correspondences both between equivalence classes of Clifford systems and those of O-systems, and between O-systems and orthogonal multiplications of the form $\mu :{ℝ}^n\times {ℝ}^m\to {ℝ}^m$, which allow us to solve the existence problems both for $O$-systems and for umbilical quadratic harmonic morphisms simultaneously. The existence problem for general quadratic harmonic morphisms is then solved by the Splitting Lemma. We also study properties possessed by all quadratic harmonic morphisms for fixed pairs of domain and range spaces.

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