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چكيده انگليسي :

harmonic morphisms and bicomplex manifols Fariba Ghazvinizadeh esfahani f ghazvinizadehesfahani@math iut ac ir August 29 2012 Master of Science Thesis Department of Mathematical Sciences Isfahan University of Technology Isfahan 84156 83111 IranSupervisor DR Mansour Aghasi m aghasi@cc iut ac irAdvisor DR Azam Etemad ae110mat@cc iut ac ir2000 MSC 58E20 53C43Key words harmonic morphism harmonic map bicomplex numberAbstractWe use functions of a bicomplex variable to unify the existing constructions of harmonic mor phisms from a 3 dimensional Euclidean space to a Riemannian or Lorentzian surface This isdone by using the notion of complex harmonic morphism between complex Riemannian manifoldsand showing how these are given by bicomplex holomorphic functions when the codomain is one bicomplex dimensional By taking real slices we recover well known compacti cations for thethree possible real cases ON the way we discuss some interesting conformal compacti cationsof complex Riemannian by interpreting them as bicomplex manifolds This led to interesting ex amples of globally de ned harmonic morphisms other than orthogonal projection and harmonicmorphisms all of whose bers are degenerate and it was shown that such degenerate harmonicmorphisms correspond to real valued null solutions of the wave equations Many notions and re sults for harmonic morphisms between semi Riemannian manifolds complexify immediately tocomplex harmonic morphisms between complex Riemannian manifolds and giving a way of con structing complex harmonic morphisms into a one dimensional bicomplex manifolds These aremany complex harmonic morphisms from open subsets of C3 to C2 B which are not obtainedby extending a real harmonic morphisms Harmonic morphisms into Riemannian or Lorentziansurfaces are particulary nice as they are conformally invariant in the sense that only the confor mal equivalence class of the metric on the codomain matters equivalently post composition ofa harmonic morphism to a surface with a weakly conformal map of surfaces is again a harmonicmorphism In particular harmonic morphism from open subsets of Minkowski 3 space into Care precisely the same as complex valud null solutions of the wave equation In the Riemanniancase they can be characterized as harmonic maps which are horizontally weakly conformal asalso called semi conformal a condition dual to weak conformality The characterization can beextened to harmonic morphisms between semi Riemannian manifolds with the additional featurethat bers can be degenerate Given bicomplex holomorphic functions q G q and q H q de ned on an open subset N of B or more generally on a one dimensional bicomplex manifolds we can form the equation 2Gz1 1 G2 z2 1 G2 z3 i2 2H and C 2 solutions q z to this equation are complex harmonic morphisms from open subsets of C3 to N and all suchharmonic morphisms which are submersive are given this way locally In general this equationde nes a congruence of lines and planes Indeed for each q N if CN G 1 this equationde nes a complex line whereas if CN G 1 either has no solutions or de nes a plane weshall call these lines and planes the ber of the congruence this equation as they form the bers ofany smooth harmonic morphism q z which satis es that equation How ever starting witharbitrary data G and H the bers of the congruence that equation may intersect or have envelopepoints where they become in nitesimally close We shall consider the behaviour of this congruencewhen the bers are degenerate or have direction not represented by a nite value of G

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