Exterior Differential Systems

Robert L. Bryant, S. S. Chern, Robert B. Gardner, Hubert L. Goldschmidt, P. A. Griffiths (auth.)9780387974118, 0387974113

This book gives a treatment of exterior differential systems. It will in­ clude both the general theory and various applications. An exterior differential system is a system of equations on a manifold defined by equating to zero a number of exterior differential forms. When all the forms are linear, it is called a pfaffian system. Our object is to study its integral manifolds, i. e. , submanifolds satisfying all the equations of the system. A fundamental fact is that every equation implies the one obtained by exterior differentiation, so that the complete set of equations associated to an exterior differential system constitutes a differential ideal in the algebra of all smooth forms. Thus the theory is coordinate-free and computations typically have an algebraic character; however, even when coordinates are used in intermediate steps, the use of exterior algebra helps to efficiently guide the computations, and as a consequence the treatment adapts well to geometrical and physical problems. A system of partial differential equations, with any number of inde­ pendent and dependent variables and involving partial derivatives of any order, can be written as an exterior differential system. In this case we are interested in integral manifolds on which certain coordinates remain independent. The corresponding notion in exterior differential systems is the independence condition: certain pfaffian forms remain linearly indepen­ dent. Partial differential equations and exterior differential systems with an independence condition are essentially the same object.

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Table of contents :
Front Matter….Pages i-vii
Introduction….Pages 1-5
Preliminaries….Pages 6-26
Basic Theorems….Pages 27-57
Cartan-Kähler Theory….Pages 58-101
Linear Differential Systems….Pages 102-169
The Characteristic Variety….Pages 170-235
Prolongation Theory….Pages 236-265
Examples….Pages 266-312
Applications of Commutative Algebra and Algebraic Geometry to the Study of Exterior Differential Systems….Pages 313-389
Partial Differential Equations….Pages 390-416
Linear Differential Operators….Pages 417-461
Back Matter….Pages 462-475