On the classification of quadratic harmonic morphisms between Euclidean spaces

Details On the classification of quadratic harmonic morphisms between Euclidean spaces On the classification of quadratic harmonic morphisms between Euclidean spaces

1995-11-03

arxiv.org/abs/dg-ga/9511002v1

Mathematics

Differential Geometry

13 pages, AMS-Latex

Scientific paper

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

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