On geometry of harmonic morphisms

ABSTRACT M. Sc. Thesis ON GEOMETRY OF HARMONIC MORPHISMS Selcen YÜKSEL İnönü University Graduate School of Natural and Applied Sciences Department of Mathematics 139+iv pages 2005 Supervisors: Prof. Dr. Sadık KELEŞ Assoc. Prof. Dr. Bayram ŞAHİN This thesis covers three chapters such a way that in the first chapter we give the basic concepts for the rest of the thesis that readers can easily understand. In this chapter we also discuss the notations and fundamental formulas required to do calculus on a Riemann manifold, Laplace's equation on a Riemann manifold and its solitions. In the second chapter weakly conformal mappings, horizontally weakly conformal mappings and conformal foliations are discussed. A harmonic morphism is a particular sort of harmonic mapping. So that in the third chapter we give some aspects of the theory of harmonic mappings which includes the second fundamental form and tension field, conservation law given by stress-energy and some basic facts on conformal minimal (branched) immersions. After then harmonic morphisms are introduced. Finally we investigated the relation between the harmonicity and horizontal weak conformal ity. KEYWORDS : Laplacian, weakly conformal map, horizontally weakly conformal map, conformal foliation, warped product, harmonic map, tension field, stress-energy tensor, minimal branched immersion, harmonic morphism. 11