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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 | |