Harmonic morphisms between Riemannian manifolds
Paul Baird and John C. Wood
London Mathematical Society Monographs, No. 29,
Oxford University Press (2003)
Description of the book
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Contents
Introduction xi
Part I Basic Facts on Harmonic Morphisms 17
- 1 Complex-valued harmonic morphisms on three-dimensional Euclidean space 3
- 1.1 Definition and characterization 3
- 1.2 Generating harmonic morphisms 6
- 1.3 A converse 9
- 1.4 Direction and displacement maps 14
- 1.5 Examples 17
- 1.6 A global theorem 21
- 1.7 Notes and comments 23
- 2 Riemannian manifolds and conformality 25
- 2.1 Riemannian manifolds 25
- 2.2 The Laplacian on a Riemannian manifold 35
- 2.3 Weakly conformal maps 40
- 2.4 Horizontally weakly conformal maps 45
- 2.5 Conformal foliations 54
- 2.6 Notes and comments 62
- 3 Harmonic mappings between Riemannian manifolds 65
- 3.1 Calculus on vector bundles 65
- 3.2 Second fundamental form and tension field 69
- 3.3 Harmonic mappings 71
- 3.4 The stress-energy tensor 81
- 3.5 Minimal branched immersions 84
- 3.6 Second variation of the energy and stability 91
- 3.7 Volume and energy 94
- 3.8 Notes and comments 100
- 4 Fundamental properties of harmonic morphisms 106
- 4.1 The Definition 106
- 4.2 Characterization 108
- 4.3 General properties 111
- 4.4 The symbol 114
- 4.5 The mean curvature of the fibres 118
- 4.6 Further consequences of the fundamental equations 124
- 4.7 Foliations which produce harmonic morphisms 128
- 4.8 Second variation 132
- 4.9 Notes and comments 136
- 5 Harmonic morphisms defined by polynomials 141
- 5.1 Entire harmonic morphisms between Euclidean spaces 141
- 5.2 Horizontally conformal polynomial maps 143
- 5.3 Orthogonal multiplications 148
- 5.4 Clifford systems 151
- 5.5 Quadratic harmonic morphisms 156
- 5.6 Homogeneous polynomial maps 162
- 5.7 Applications to horizontally weakly conformal maps 167
- 5.8 Notes and comments 169
Part II Twistor Methods 173
- 6 Mini-twistor theory on three-dimensional space forms 175
- 6.1 Factorization of harmonic morphisms from 3-manifolds 175
- 6.2 Geodesics on a three-dimensional space form 180
- 6.3 The space of oriented geodesics on Euclidean 3-space 183
- 6.4 The space of oriented geodesics on the 3-sphere 185
- 6.5 The space of oriented geodesics on hyperbolic 3-space 188
- 6.6 Harmonic morphisms from three-dimensional space forms 189
- 6.7 Entire harmonic morphisms on space forms 194
- 6.8 Higher dimensions 199
- 6.9 Notes and comments 203
- 7 Twistor methods 206
- 7.1 The twistor space of a Riemannian manifold 206
- 7.2 Kählerian twistor spaces 211
- 7.3 The twistor space of the 4-sphere 214
- 7.4 The twistor space of Euclidean 4-space 216
- 7.5 The twistor spaces of complex projective 2-space 217
- 7.6 The twistor space of an anti-self-dual 4-manifold 219
- 7.7 Adapted Hermitian structures 220
- 7.8 Superminimal surfaces 223
- 7.9 Hermitian structures from harmonic morphisms 228
- 7.10 Harmonic morphisms from Hermitian structures 231
- 7.11 Harmonic morphisms from Euclidean 4-space 236
- 7.12 Harmonic morphisms from the 4-sphere 239
- 7.13 Harmonic morphisms from complex projective 2-space 241
- 7.14 Harmonic morphisms from other Einstein 4-manifolds 243
- 7.15 Notes and comments 244
- 8 Holomorphic harmonic morphisms 250
- 8.1 Harmonic morphisms between almost Hermitian manifolds 250
- 8.2 Composition laws 254
- 8.3 Hermitian structures on open subsets of Euclidean spaces 257
- 8.4 The Weierstrass formulae 259
- 8.5 Reduction to odd dimensions and to spheres 262
- 8.6 General holomorphic harmonic morphisms on Euclidean spaces 266
- 8.7 Notes and comments 270
- 9 Multivalued harmonic morphisms 273
- 9.1 Multivalued mappings 274
- 9.2 Multivalued harmonic morphisms 276
- 9.3 Classes of Examples 281
- 9.4 An alternative treatment for space forms 283
- 9.5 Some specific examples 284
- 9.6 Behaviour on the branching set 288
- 9.7 Notes and comments 292
Part III Topological and Curvature Considerations 293
- 10 Harmonic morphisms from compact 3-manifolds 295
- 10.1 Seifert fibre spaces 295
- 10.2 Three-dimensional geometries 300
- 10.3 Harmonic morphisms and Seifert fibre spaces 302
- 10.4 Examples 305
- 10.5 Characterization of the metric 307
- 10.6 Propagation of fundamental quantities along the fibres 312
- 10.7 Notes and comments 317
- 11 Curvature considerations 319
- 11.1 The fundamental tensors 319
- 11.2 Curvature for a horizontally conformal submersion 320
- 11.3 Walczak's formula 327
- 11.4 Conformal maps between equidimensional manifolds 330
- 11.5 Curvature and harmonic morphisms 332
- 11.6 Weitzenböck formulae 338
- 11.7 Curvature for one-dimensional fibres 341
- 11.8 Entire harmonic morphisms on Euclidean space with totally geodesic fibres 347
- 11.9 Notes and comments 349
- 12 Harmonic morphisms with one-dimensional fibres 352
- 12.1 Topological restrictions 352
- 12.2 The normal form of the metric 360
- 12.3 Harmonic morphisms of Killing type 364
- 12.4 Harmonic morphisms of warped product type 366
- 12.5 Harmonic morphisms of type (T) 371
- 12.6 Uniqueness of types 374
- 12.7 Einstein manifolds 375
- 12.8 Harmonic morphisms from an Einstein 4-manifold 378
- 12.9 Constant curvature manifolds 383
- 12.10 Notes and comments 389
- 13 Reduction techniques 392
- 13.1 Isoparametric mappings 392
- 13.2 Eigen-harmonic morphisms 398
- 13.3 Reduction 399
- 13.4 Conformal changes of the metrics 402
- 13.5 Reduction to an ordinary differential equation 405
- 13.6 Reduction to a partial differential equation 413
- 13.7 Notes and comments 419
Part IV Further Developments 425
- 14 Harmonic morphisms between semi-Riemannian manifolds 427
- 14.1 Semi-Riemannian manifolds 427
- 14.2 Harmonic maps between semi-Riemannian manifolds 435
- 14.3 Harmonic maps between Lorentzian surfaces 438
- 14.4 Weakly conformal maps and stress-energy 440
- 14.5 Horizontally weakly conformal maps 444
- 14.6 Harmonic morphisms between semi-Riemannian manifolds 446
- 14.7 Harmonic morphisms between Lorentzian surfaces 449
- 14.8 Notes and comments 452
- Appendix 456
- A.1 Analytic aspects of harmonic functions 456
- A.2 A regularity result for an equation of Yamabe type 460
- A.3 A technical result on the symbol 462
- A.4 Notes and comments 465
References 467
Glossary of notation 499
Index 502