Transformations of Manifolds and Applications to Differential Equations: (Monographs and Surveys in Pure and Applied Mathematics)

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.

Table of Contents:
Transformations of Surfaces and Applications The Structure Equations Differential Equations Associated to Linear Weingarten Surfaces Geodesic Congruences and Parallel Surfaces Pseudo-Spherical Geodesic Congruences Bäcklund 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 Bäcklund's Theorem Permutability Theorem Bäcklund Transformation and Superposition Formula for the Generalized Wave Equation and the Generalized sine-Gordon Equation Linearization of the Bäcklund Transformation The Inverse Scattering Method for the Generalized Wave Equation The Inverse Scattering Method for the Generalized sine-Gordon Equation The Bäcklund Transformation in Terms of Scattering Data. Soliton Solutions The Generating Equation The Generating Equation Bäcklund 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 Bäcklund 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