Harmonic Morphisms beteen Riemannian Manifolds [Hardcover]

This is the first account in book form of the theory of harmonic morphisms between Riemannian manifolds. Harmonic morphisms are maps which preserve Laplace's equation. They can be characterized as harmonic maps which satisfy an additional first order condition. Examples include harmonic functions, conformal mappings in the plane, and holomorphic functions with values in a Riemann surface. There are connections with many concepts in differential geometry, for example, Killing fields, geodesics, foliations, Clifford systems, twistor spaces, Hermitian structures, iso-parametric mappings, and Einstein metrics and also the Brownain pathpreserving maps of probability theory. Giving a complete account of the fundamental aspects of the subject, this book is self-contained, assuming only a basic knowledge of differential geometry.

Introduction
IBasic Facts on Harmonic Morphisms
1. Complex-valued harmonic morphisms on 3-dimensional Euclidean space
2. Riemannian manifolds and conformality
3. Harmonic mappings between Riemannian manifolds
4. Fundamental properties of harmonic morphisms
5. Harmonic morphisms defined by polynomials
IITwistor Methods
6. Mini-twistor theory on 3-dimensional space-forms
7. Twistor methods
8. Holomorphic harmonic morphisms
9. Multivalued harmonic morphisms
IIITopological and Curvature considerations
10. Harmonic morphisms for compact 3-manifolds
11. Curvature considerations
12. Harmonic morphisms with one-dimensional fibres
13. Reduction techniques
IVFurther Developments
14. Harmonic morphisms between semi-Riemannian manifolds
Appendix
Bibliography
Index