Huygens' Principle and Hyperbolic Equations

Contents 949
Preface 953
Acknowledgements 961
Introduction 963
Chapter I 1
 §1 Normal domains 1
 §2 The causal structure of space-times 8
 §3 Vector bundles 20
 §4 The wave equations for differential forms in non-euclidean spaces 27
 §5 A spinor calculus 64
Chapter II Riesz distributions 105
 §1 The Riesz distributions in the Minkowski space 105
 §2 The Riesz distributions in curved space-times 119
 §3 Some generalizations 140
Chapter III The fundamental solutions 153
 §1 The Hadamard coefficients 153
 §2 B-series 164
 §3 The fundamental solutions 174
 §4 Applications of the fundamental solutions 201
 §5 The Cauchy problem 213
Chapter IV Huygens' operators 229
 §1 Hadamard's criterion 229
 §2 Huygens' triples 249
 §3 Diversors. General wave families 266
 §4 Maxwell's equations. Dirac's equations 281
Chapter V The Euler-Poisson-Darboux equation 303
 §1 An application of the method of descent 303
 §2 The singular Cauchy problem 316
 §3 Huygens' principle for the EPD-equation 342
 §4 Stellmacher's equations 355
 §5 Elllptic operators with vanishing flrst Hadamard coefficient 389
  Appendix 421
 §6 Relations to spectral geometry 434
Chapter VI Transformation theory 459
 §1 The bundle connection associated to an operator P 459
 §2 A property of the Hadamard coefficients 471
 §3 Conformal gauge transformations of an operator P 479
 §4 Tensors with simple transformation law 495
 §5 The moments of a normal hyperbolic operator (even dimension) 518
 §6 The moments for Maxwell's equations 533
Chapter VII Some theorems on Huygens' operators over four-dimensional space-times 555
 §1 Some preparatory transformations 555
 §2 The moments of order <= 3 567
 §3 Applications to Huygens' operators in a four-dimensional space-time 587
 §4 The case of conformally flat metrics 608
Chapter VIII Plane wave manifolds and Huygens' principle 641
 §1 Introduction. Results 641
 §2 pp- and plane wave manifolds 650
 §3 Huygens' principle for plane wave manifolds 670
 §4 A characterization of plane wave manifolds 692
 §5 Some conformally invariant tensors 718
 §6 Testing coefficients by pp-metrics 745
 §7 Testing coefficients by metrics of constant curvature 765
Table I Identities for the Weyl tensor 795
Table II Moments of order <= 4 in four dimensions 799
Table III Some formulas for pp-metrics 803
Table IV Some formulas for plane wave metrics 805
Appendix I Metric and curvature in normal coordinates 807
Appendix II Weak Huygens' operators by V. Wünsch 821
Appendix III Huygens' principle for spin tensor equations by V. Wünsch 825
Index 831
Bibliography 833