Harmonic morphisms and bicomplex manifolds

Details Harmonic morphisms and bicomplex manifolds Harmonic morphisms and bicomplex manifolds

2009-10-06

arxiv.org/abs/0910.1036v2

Mathematics

Differential Geometry

29 pages. Previously called `Harmonic morphisms and bicomplex numbers'; minor improvements made.

Scientific paper

We use functions of a bicomplex variable to unify the existing constructions of harmonic morphisms from a 3-dimensional Euclidean or pseudo-Euclidean space to a Riemannian or Lorentzian surface. This is done by using the notion of complex-harmonic morphism between complex-Riemannian manifolds and showing how these are given by bicomplex-holomorphic functions when the codomain is one-bicomplex dimensional. By taking real slices, we recover well-known compactifications for the three possible real cases. On the way, we discuss some interesting conformal compactifications of complex-Riemannian manifolds by interpreting them as bicomplex manifolds.

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