The study of the interaction between differential geometry and partial differential equations has a long history-dating to the last century-and continues to generate considerable interest. Most of the local properties of manifolds are expressed in terms of partial differential equations, and this correspondence proves useful in two ways: we can obtain solutions to the equations from our knowledge about the local geometry of the manifolds, and we can obtain geometric properties of the manifolds-or even prove the non-existence of certain geometric structures on manifolds-from our knowledge of the differential equations.
Transformations of Manifolds and Applications to Differential Equations focuses on the role played by differential geometry on the study of differential equations. The author combines the geometric and analytic aspects of the theory, not only in the classical examples, but also in more recent results on integrable systems with an arbitrary number of independent variables.
With its applications to problems in evolution equations, strongly hyperbolic systems of the hydrodynamic type, linear Weingarten surfaces, and submanifolds of constant curvature, this volume will prove interesting and valuable to researchers and mathematicians working in differential geometry, differential equations, and mathematical physics.
Reihe
Sprache
Verlagsort
Verlagsgruppe
Zielgruppe
Für höhere Schule und Studium
Für Beruf und Forschung
Mathematicians, graduate students, and researchers in differential geometry, differential equations, and mathematical physics
Mathematical physicists
Gewicht
ISBN-13
978-1-58488-034-9 (9781584880349)
Copyright in bibliographic data is held by Nielsen Book Services Limited or its licensors: all rights reserved.
Schweitzer Klassifikation
Transformations of Surfaces and Applications
The Structure Equations
Differential Equations Associated to Linear Weingarten Surfaces
Geodesic Congruences and Parallel Surfaces
Pseudo-Spherical Geodesic Congruences
Baecklund Transformation for the sine-Gordon and the Elliptic sinh-Gordon Equations. Superposition Formula
The Laplace Transformation for Second-Order Hyperbolic Equations and its Geometric Interpretation
Differential Equations which Describe Pseudo-Spherical Surfaces
Submanifolds of Constant Sectional Curvature
The Structure Equations in a Pseudo-Riemannian Space Form
Submanifolds of Constant Sectional Curvature. The Generating Equation
Pseudo-Spherical Geodesic Congruences and Applications
Pseudo-Spherical Geodesic Congruences. A Generalization of Baecklund's Theorem
Permutability Theorem
Baecklund Transformation and Superposition Formula for the Generalized Wave Equation and the Generalized sine-Gordon Equation
Linearization of the Baecklund Transformation
The Inverse Scattering Method for the Generalized Wave Equation
The Inverse Scattering Method for the Generalized sine-Gordon Equation
The Baecklund Transformation in Terms of Scattering Data. Soliton Solutions
The Generating Equation
The Generating Equation
Baecklund Transformation for the Generating Equation and its Linearization
Superposition Formula
The Generating Intrinsic Equation
The Generating Intrinsic Equation. Submanifolds of Constant Curvature Characterized by the Metric
Baecklund Transformation for the Generating Intrinsic Equation. Symmetry Group
Hyperbolic Toroidal Submanifolds of Euclidean Space
Flat Toroidal Submanifolds of the Unit Sphere
Geometric Properties of Submanifolds Associated to Special Solutions
Laplace Transformation in Higher Dimensions
Laplace Transformations of Cartan Manifolds
The Higher-Dimensional Laplace Invariants for Systems of Second Order PDEs
The Generalized Method of Laplace for Systems of Second Order PDEs
Applications of the Laplace Transformation to Hydrodynamic Systems Rich in Conservation Laws
References
Index
