2009/10/06 by Paul Baird, Baird, Paul, John C. Wood +1
Mathematics · #53C43 #58E20 #Differential Geometry (math.DG) #FOS: Mathematics #math.DG #msc:53C43 #msc:58E20
paper · pdf · doi:10.48550/arxiv.0910.1036
29 pages. Previously called `Harmonic morphisms and bicomplex numbers'; minor improvements made.
arxiv created 2010/03/11 · arxiv updated 2010/03/12
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.