Hurwitz problem, harmonic morphisms and generalized Hadamard matrices
The authors use real representations of real Clifford algebras associated with the Hurwitz problem of factorization of algebraic sums of squares in order to construct harmonic morphisms defined on Euclidean spaces or on pseudo-Euclidean spaces endowed with a metric of neutral signature. In general the composition of two harmonic maps is not a harmonic map. But the composition of a harmonic morphism with a harmonic map is also harmonic. The result of the paper is an extension of the construction given by \textit{P. Baird} [Harmonic maps with symmetric, harmonic morphisms and deformations of metrics (1983; Zbl 0515.58010)] in the framework of the theory of harmonic maps defined on Euclidean spaces. The construction of harmonic morphisms considered in the paper can be used to generate generalized Hadamard matrices which are used in design theory. Moreover, on the base of the Hurwitz transformations generalizing the Kustaanheimo-Steifel transformations the authors give a way to construct a class of harmonic functions on \(\mathbb{R}^{m+1}\) from those of \(\mathbb{R}^{s+ 1}\) and conversely.
scientific article; zbMATH DE number 739558
scientific article