Exterior Differential Systems: (18 Mathematical Sciences Research Institute Publications)

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.

Table of Contents:
I. Preliminaries.- §1. Review of Exterior Algebra.- §2. The Notion of an Exterior Differential System.- §3. Jet Bundles.- II. Basic Theorems.- §1. Probenius Theorem.- §2. Cauchy Characteristics.- §3. Theorems of Pfaff and Darboux.- §4. Pfaffian Systems.- §5. Pfaffian Systems of Codimension Two.- III. Cartan-Kähler Theory.- §1. Integral Elements.- §2. The Cartan-Kähler Theorem.- §3. Examples.- IV. Linear Differential Systems.- §1. Independence Condition and Involution.- §2. Linear Differential Systems.- §3. Tableaux.- §4. Tableaux Associated to an Integral Element.- §5. Linear Pfaffian Systems.- §6. Prolongation.- §7. Examples.- §8. Families of Isometric Surfaces in Euclidean Space.- V. The Characteristic Variety.- §1. Definition of the Characteristic Variety of a Differential System.- §2. The Characteristic Variety for Linearc Pfaffian Systems; Examples.- §3. Properties of the Characteristic Variety.- VI. Prolongation Theory.- §1. The Notion of Prolongation.- §2. Ordinary Prolongation.- §3. The Prolongation Theorem.- §4. The Process of Prolongation.- VII. Examples.- §1. First Order Equations for Two Functions of Two Variables.- §2. Finiteness of the Web Rank.- §3. Orthogonal Coordinates.- §4. Isometric Embedding.- VIII. Applications of Commutative Algebra and Algebraic Geometry to the Study of Exterior Differential Systems.- §1. Involutive Tableaux.- §2. The Cartan-Poincaré Lemma, Spencer Cohomology.- §3. The Graded Module Associated to a Tableau; Koszul Homology.- §4. The Canonical Resolution of an Involutive Module.- §5. Localization; the Proofs of Theorem 3.2 and Proposition 3.8.- §6. Proof of Theorem 3.8 in Chapter V; Guillemin’s Normal Form.- §7. The Graded Module Associated to a Higher Order Tableau.- IX. PartialDifferential Equations.- §1. An Integrability Criterion.- §2. Quasi-Linear Equations.- §3. Existence Theorems.- X. Linear Differential Operators.- §1. Formal Theory and Complexes.- §2. Examples.- §3. Existence Theorems for Elliptic Equations.