Harmonic Morphisms Between Riemannian Manifolds
Paul Baird and John C. Wood
Abstract
Harmonic morphisms are maps which preserve Laplace's equation. More explicitly, a map between Riemannian manifolds is called a harmonic morphism if its composition with any locally defined harmonic function on the codomain is a harmonic function on the domain; it thus ‘pulls back’ germs of harmonic functions to germs of harmonic functions. Harmonic morphisms can be characterized as harmonic maps satisfying a condition dual to weak conformality called ‘horizontal weak conformality’ or ‘semiconformality’. Examples include harmonic functions, conformal mappings in the plane, holomorphic mappings ... More
Harmonic morphisms are maps which preserve Laplace's equation. More explicitly, a map between Riemannian manifolds is called a harmonic morphism if its composition with any locally defined harmonic function on the codomain is a harmonic function on the domain; it thus ‘pulls back’ germs of harmonic functions to germs of harmonic functions. Harmonic morphisms can be characterized as harmonic maps satisfying a condition dual to weak conformality called ‘horizontal weak conformality’ or ‘semiconformality’. Examples include harmonic functions, conformal mappings in the plane, holomorphic mappings with values in a Riemann surface, and certain submersions arising from Killing fields and geodesic fields. The study of harmonic morphisms involves many different branches of mathematics: the book includes discussion on aspects of the theory of foliations, polynomials induced by Clifford systems and orthogonal multiplications, twistor and mini-twistor spaces, and Hermitian structures. Relations with topology are discussed, including Seifert fibre spaces and circle actions, also relations with isoparametric functions and the Beltrami fields equation of hydrodynamics.
Keywords:
harmonic map,
Laplace's equation,
conformal,
Killing field,
twistor,
foliation
Bibliographic Information
| Print publication date: 2003 |
Print ISBN-13: 9780198503620 |
| Published to Oxford Scholarship Online: September 2007 |
DOI:10.1093/acprof:oso/9780198503620.001.0001 |
Authors
Affiliations are at time of print publication.
Paul Baird, author
Professeur de Mathématiques, Université de Bretagne Occidentale, Brest
John C. Wood, author
Professor of Pure Mathematics, University of Leeds
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