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| Introduction | |
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| Basic Facts on Harmonic Morphisms | |
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| Complex-valued harmonic morphisms on three-dimensional Euclidean space | |
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| Definition and characterization | |
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| Generating harmonic morphisms | |
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| A converse | |
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| Direction and displacement maps | |
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| Examples | |
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| A global theorem | |
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| Notes and comments | |
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| Riemannian manifolds and conformality | |
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| Riemannian manifolds | |
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| The Laplacian on a Riemannian manifold | |
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| Weakly conformal maps | |
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| Horizontally weakly conformal maps | |
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| Conformal foliations | |
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| Notes and comments | |
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| Harmonic mappings between Riemannian manifolds | |
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| Calculus on vector bundles | |
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| Second fundamental form and tension field | |
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| Harmonic mappings | |
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| The stress-energy tensor | |
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| Minimal branched immersions | |
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| Second variation of the energy and stability | |
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| Volume and energy | |
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| Notes and comments | |
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| Fundamental properties of harmonic morphisms | |
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| The Definition | |
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| Characterization | |
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| General properties | |
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| The symbol | |
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| The mean curvature of the fibres | |
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| Further consequences of the fundamental equations | |
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| Foliations which produce harmonic morphisms | |
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| Second variation | |
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| Notes and comments | |
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| Harmonic morphisms defined by polynomials | |
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| Entire harmonic morphisms between Euclidean spaces | |
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| Horizontally conformal polynomial maps | |
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| Orthogonal multiplications | |
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| Clifford systems | |
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| Quadratic harmonic morphisms | |
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| Homogeneous polynomial maps | |
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| Applications to horizontally weakly conformal maps | |
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| Notes and comments | |
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| Twistor Methods | |
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| Mini-twistor theory on three-dimensional space forms | |
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| Factorization of harmonic morphisms from 3-manifolds | |
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| Geodesics on a three-dimensional space form | |
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| The space of oriented geodesics on Euclidean 3-space | |
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| The space of oriented geodesics on the 3-sphere | |
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| The space of oriented geodesics on hyperbolic 3-space | |
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| Harmonic morphisms from three-dimensional space forms | |
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| Entire harmonic morphisms on space forms | |
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| Higher dimensions | |
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| Notes and comments | |
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| Twistor methods | |
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| The twistor space of a Riemannian manifold | |
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| Kahlerian twistor spaces | |
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| The twistor space of the 4-sphere | |
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| The twistor space of Euclidean 4-space | |
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| The twistor spaces of complex projective 2-space | |
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| The twistor space of an anti-self-dual 4-manifold | |
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| Adapted Hermitian structures | |
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| Superminimal surfaces | |
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| Hermitian structures from harmonic morphisms | |
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| Harmonic morphisms from Hermitian structures | |
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| Harmonic morphisms from Euclidean 4-space | |
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| Harmonic morphisms from the 4-sphere | |
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| Harmonic morphisms from complex projective 2-space | |
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| Harmonic morphisms from other Einstein 4-manifolds | |
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| Notes and comments | |
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| Holomorphic harmonic morphisms | |
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| Harmonic morphisms between almost Hermitian manifolds | |
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| Composition laws | |
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| Hermitian structures on open subsets of Euclidean spaces | |
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| The Weierstrass formulae | |
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| Reduction to odd dimensions and to spheres | |
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| General holomorphic harmonic morphisms on Euclidean spaces | |
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| Notes and comments | |
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| Multivalued harmonic morphisms | |
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| Multivalued mappings | |
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| Multivalued harmonic morphisms | |
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| Classes of Examples | |
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| An alternative treatment for space forms | |
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| Some specific examples | |
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| Behaviour on the branching set | |
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| Notes and comments | |
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| Topological and Curvature Considerations | |
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| Harmonic morphisms from compact 3-manifolds | |
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| Seifert fibre spaces | |
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| Three-dimensional geometries | |
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| Harmonic morphisms and Seifert fibre spaces | |
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| Examples | |
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| Characterization of the metric | |
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| Propagation of fundamental quantities along the fibres | |
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| Notes and comments | |
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| Curvature considerations | |
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| The fundamental tensors | |
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| Curvature for a horizontally conformal submersion | |
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| Walczak's formula | |
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| Conformal maps between equidimensional manifolds | |
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| Curvature and harmonic morphisms | |
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| Weitzenbock formulae | |
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| Curvature for one-dimensional fibres | |
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| Entire harmonic morphisms on Euclidean space with totally geodesic fibres | |
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| Notes and comments | |
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| Harmonic morphisms with one-dimensional fibres | |
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| Topological restrictions | |
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| The normal form of the metric | |
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| Harmonic morphisms of Killing type | |
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| Harmonic morphisms of warped product type | |
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| Harmonic morphisms of type (T) | |
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| Uniqueness of types | |
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| Einstein manifolds | |
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| Harmonic morphisms from an Einstein 4-manifold | |
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| Constant curvature manifolds | |
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| Notes and comments | |
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| Reduction techniques | |
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| Isoparametric mappings | |
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| Eigen-harmonic morphisms | |
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| Reduction | |
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| Conformal changes of the metrics | |
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| Reduction to an ordinary differential equation | |
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| Reduction to a partial differential equation | |
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| Notes and comments | |
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| Further Developments | |
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| Harmonic morphisms between semi-Riemannian manifolds | |
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| Semi-Riemannian manifolds | |
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| Harmonic maps between semi-Riemannian manifolds | |
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| Harmonic maps between Lorentzian surfaces | |
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| Weakly conformal maps and stress-energy | |
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| Horizontally weakly conformal maps | |
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| Harmonic morphisms between semi-Riemannian manifolds | |
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| Harmonic morphisms between Lorentzian surfaces | |
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| Notes and comments | |
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| Appendix | |
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| Analytic aspects of harmonic functions | |
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| A regularity result for an equation of Yamabe type | |
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| A technical result on the symbol | |
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| Notes and comments | |
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| References | |
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| Glossary of notation | |
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| Index | |