Harmonic Maps and Morphisms from Multilinear Norm-Preserving Mappings

Abstract

We extend the notion of orthogonal multiplication to multilinear norm-preserving mapping, using them to construct new eigenmaps into spheres. We characterize those which are harmonic morphisms. By the method of reduction we construct interesting families of harmonic morphisms into S2 from the product manifolds H2 × S3 and S3 × S3 of hyperbolic spaces and spheres. The corresponding reduction equation depends on two independent variables. We are able to solve the first-order horizontal conformality problem explicitly in terms of elliptic functions and then render the map harmonic by a conformal deformation of the metric.