Huygens' Principle and Hyperbolic Equations

Preface

Acknowledgements

Introduction

Chapter I

§1 Normal Domains

§2 The Causal Structure of Space-Times

§3 Vector Bundles

§4 The Wave Equations for Differential Forms in Non-Euclidean Spaces

§5 A Spinor Calculus

Chapter II Riesz Distributions

§1 The Riesz Distributions in the Minkowski Space

§2 The Riesz Distributions in Curved Space-Times

§3 Some Generalizations

Chapter III The Fundamental Solutions

§1 The Hadamard Coefficients

§2 B-Series

§3 The Fundamental Solutions

§4 Applications of the Fundamental Solutions

§5 The Cauchy Problem

Chapter IV Huygens' Operators

§1 Hadamard's Criterion

§2 Huygens' Triples

§3 Diversors. General Wave Families

§4 Maxwell's Equations. Dirac's Equations

Chapter V The Euler-Poisson-Darboux Equation

§1 An Application of the Method of Descent

§2 The Singular Cauchy Problem

§3 Huygens' Principle for the EPD-Equation

§4 Stellmacher's Equations

§5 Elliptic Operators with Vanishing First Hadamard Coefficient

Appendix

§6 Relations to Spectral Geometry

Chapter VI Transformation Theory

§1 The Bundle Connection Associated to an Operator P

§2 A Property of the Hadamard Coefficients

§3 Conformal Gauge Transformations of an Operator P

§4 Tensors with Simple Transformation Law

§5 The Moments of a Normal Hyperbolic Operator (Even Dimension)

§6 The Moments for Maxwell's Equations

Chapter VII Some Theorems on Huygens' Operators Over Four-Dimensional Space-Times

§1 Some Preparatory Transformations

§2 The Moments of Order ≤ 3

§3 Applications to Huygens' Operators in a Four-Dimensional Space-Time

§4 The Case of Conformally Flat Metrics

Chapter VIII Plane Wave Manifolds and Huygens' Principle

§1 Introduction. Results

§2 pp- and Plane Wave Manifolds

§3 Huygens' Principle for Plane Wave Manifolds

§4 A Characterization of Plane Wave Manifolds

§5 Some Conformally Invariant Tensors

§6 Testing Coefficients by pp-Metrics

§7 Testing Coefficients by Metrics of Constant Curvature

Table I Identities for the Weyl Tensor

Table II Moments of Order ≤ 4 in Four Dimensions

Table III Some Formulas for pp-Metrics

Table IV Some Formulas for Plane Wave Metrics

Appendix I Metric and Curvature in Normal Coordinates

Appendix II Weak Huygens' Operators by V. Wünsch

Appendix III Huygens' Principle for Spin Tensor Equations by V. Wünsch

Index

Bibliography