Harmonic morphisms between Riemannian manifolds / Paul Baird and John C. Wood.

Format:

Book

Author/Creator:

Baird, P. (Paul)

Contributor:

Wood, John C.

Series:

London Mathematical Society monographs ; new ser., v. 29.

London Mathematical Society monographs ; new ser., v. 29

Language:

English

Subjects (All):

Riemannian manifolds.

Harmonic morphisms.

Physical Description:

xvi, 520 pages : illustrations ; 25 cm.

Place of Publication:

Oxford : Clarendon ; New York : Oxford university press, 2003.

Summary:

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, iso-parametric mappings, and Einstein metrics and also the Brownain pathpreserving 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.

Contents:

I Basic Facts on Harmonic Morphisms

1 Complex-valued harmonic morphisms on three-dimensional Euclidean space 3

1.2 Generating harmonic morphisms 6

1.3 A converse 9

1.4 Direction and displacement maps 14

1.6 A global theorem 21

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

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

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

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

II Twistor Methods

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

7 Twistor methods 206

7.1 The twistor space of a Riemannian manifold 206

7.2 Kahlerian 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

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

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

III Topological and Curvature Considerations

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

10.5 Characterization of the metric 307

10.6 Propagation of fundamental quantities along the fibres 312

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 Weitzenbock formulae 338

11.7 Curvature for one-dimensional fibres 341

11.8 Entire harmonic morphisms on Euclidean space with totally geodesic fibres 347

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

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

IV Further Developments

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

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.

Notes:

Includes bibliographical references and index.

ISBN:

0198503628

OCLC:

51316121