Huygens' Principle and Hyperbolic Equations - 1st Edition - ISBN: 9780123073303, 9781483262222

Huygens' Principle and Hyperbolic Equations, Volume 5

1st Edition

Editors: J. Coates S. Helgason
Authors: Gunther Paul
eBook ISBN: 9781483262222
Imprint: Academic Press
Published Date: 28th May 1988
Page Count: 906
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Table of Contents




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


§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




Huygens' Principle and Hyperbolic Equations is devoted to certain mathematical aspects of wave propagation in curved space-times.

The book aims to present special nontrivial Huygens' operators and to describe their individual properties and to characterize these examples of Huygens' operators within certain more or less comprehensive classes of general hyperbolic operators. The materials covered in the book include a treatment of the wave equation for p-forms over a space of constant sectional curvature, the Riesz distributions, the Euler-Poisson-Darboux-equations over a Riemannian manifold, and plane wave manifolds.

Physicists will find the book invaluable.


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© Academic Press 1988
Academic Press
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Ratings and Reviews

About the Editors

J. Coates Editor

S. Helgason Editor

About the Authors

Gunther Paul Author