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, isoparametric mappings, and Einstein metrics, and also the Brownian path-preserving 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. One chapter follows the complete development of the fundamental geometric aspects of harmonic maps from scratch.This text is suitable for a beginning graduate student interested in harmonic maps and morphisms, or related subjects. The student is brought to the frontiers of
knowledge in this rapidly expanding field in which there are many interesting avenues of research to be developed.The authors are world leaders in the field, and have established
many of the key results. In this book they have brought together their work and the work of many others to form a coherent account of the subject.This book is the 29th volume in the London Mathematical Society Monographs series, published by Oxford University press on behalf of the London Mathematical Society. The series contains authoritative accounts of current research in mathematics and high quality expository works bringing the reader to the frontiers of
research. Of particular interest are topics that have developed rapidly in the past ten years or so, but which have reached a certain level of maturity. Clarity of exposition is important and each book
contains preliminary material to make the topic accessible to those commencing work in this area.
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, isoparametric mappings, and Einstein metrics, and also the Brownian path-preserving 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. One chapter follows the complete development of the fundamental geometric aspects of harmonic maps from scratch.This text is suitable for a beginning graduate student interested in harmonic maps and morphisms, or related subjects. The student is brought to the frontiers of
knowledge in this rapidly expanding field in which there are many interesting avenues of research to be developed.The authors are world leaders in the field, and have established
many of the key results. In this book they have brought together their work and the work of many others to form a coherent account of the subject.This book is the 29th volume in the London Mathematical Society Monographs series, published by Oxford University press on behalf of the London Mathematical Society. The series contains authoritative accounts of current research in mathematics and high quality expository works bringing the reader to the frontiers of
research. Of particular interest are topics that have developed rapidly in the past ten years or so, but which have reached a certain level of maturity. Clarity of exposition is important and each book
contains preliminary material to make the topic accessible to those commencing work in this area.
Introduction
IBasic Facts on Harmonic Morphisms
1: Complex-valued harmonic morphisms on three-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 three-dimensional space-forms
7: Twistor methods
8: Holomorphic harmonic morphisms
9: Multivalued harmonic morphisms
IIITopological and Curvature considerations
10: Harmonic morphisms from 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
Glossary of Notation
Bibliography
Index
The book is written by two of the foremost experts on harmonic maps and harmonic morphisms. Serious dedication and commitment to the quality and scope of the work have resulted in this veritable opus. The exposition is lucid and authorative, making it a highly enjoyable reading, as well as a powerful reference tool. Bulletin London Math Society Vol 38, 2006 This informative and inspiring book gathers the most important results on harmonic morpisms into a single volume, presenting them in a unified and modern way. Sigmundur Gudmundsson and Martin Svensson, Finite Packing and Covering
|
Ask a Question About this Product More... |
|
