##
**Exterior differential systems.**
*(English)*
Zbl 0726.58002

Mathematical Sciences Research Institute Publications 18. New York etc.: Springer-Verlag (ISBN 0-387-97411-3). vii, 475 p. (1991).

The general theory of exterior differential systems was created by E. Cartan in connection with the theory of Lie pseudogroups and the moving frame method. This theory, which is equivalent to the theory of systems of partial differential equations, is of fundamental importance for solving the existence problems in differential geometry. On the other hand, Spencer’s theory of overdetermined systems of partial differential equations originated in the deformation theory of pseudogroup structures. The basic advantage of Spencer’s methods is that they enable us to study systems of class \(C^{\infty}\), while Cartan’s approach is properly related with the real analytic case. The book under review treats both theories and presents many geometric applications.

Chapter I gives a review of exterior algebra and an introduction to jet bundles. An exterior differential system on a manifold M is defined as an ideal \({\mathcal I}\) in the exterior algebra of M that is closed under exterior differentiation. Usually one prescribes an independence condition for \({\mathcal I}\) in the form of a decomposable exterior p-form \(\Omega\) on M. Then the integral manifolds of (\({\mathcal I},\Omega)\) are the p-dimensional integral manifolds N of \({\mathcal I}\) satisfying \(\Omega |_ N\neq 0\). Chapter II deals with simple exterior differential systems that can be studied using ordinary differential equations. First the classical results by Frobenius, Pfaff, Darboux and others are deduced. Then the Bryant normal form of some Pfaffian systems (generated by 1-forms) is derived.

Chapter III presents the Cartan-Kähler existence theorem, which is based on repeated application of the Cauchy-Kowalewski theorem, so that all data are assumed to be real analytic. The proof follows the classical one by Kähler, but the presentation is extremely clear owing to numerous methodical improvements by the authors. In particular, the Cartan test for an ordinary integral element is clarified from several points of view. Then some geometric applications are discussed, the most important of which is the proof of the Cartan-Janet theorem on local isometric embedding of a Riemannian n-space into the Euclidean \(n(n+1)/2\)-space. The main goal of Chapter IV is to develop the formalism of linear Pfaffian systems in a form that facilitates the concrete computation in geometric problems. An arbitrary system (\({\mathcal I},\Omega)\) is said to be in involution at \(x\in M\), if there exists an ordinary integral element \(E\subset T_ xM\). The most important algebraic problems from the theory of systems in involution are reflected in the concept of a tableau. In general, a tableau is a linear subspace \(A\subset Hom(V,W)\), where V and W are vector spaces. An involutive tableau is defined purely algebraically and it is deduced that the tableau \(A_ E\) of the linearization (\({\mathcal I}_ E,\Omega_ E)\) of (\({\mathcal I},\Omega)\) at E is involutive at every ordinary integral element E. Then the linear Pfaffian systems (linear with respect to a natural affine structure) are discussed in detail. Such a system can be determined by two subbundles \(I\subset J\subset T^*M\) satisfying \(dI=0 mod\{J\}\). This setting leads directly to the concept of the torsion of (I,J). A very practical result is that a linear Pfaffian system is in involution at \(x\in M\) if and only if its torsion vanishes locally and its tableau at x is involutive. This theorem is applied to solve several geometric problems. Chapter V is devoted to the characteristic variety of an exterior differential system. Several examples illustrate its important role in geometry.

A classical conjecture by E. Cartan reads that after a finite number of prolongations every differential system becomes involutive (systems with no solutions being allowed as a special case). Precise and effective results of this type represent a complicated problem (some known results are based on certain assumptions that can be verified in the course of the prolongation procedure only). That is why the authors present a subtle definition of a prolongation sequence of an exterior differential system in Chapter VI. Then a version of the Cartan-Kuranishi prolongation theorem is proved. The best results can be deduced for a linear Pfaffian system, the tableau of which is 2-acyclic in the sense of Chapter VIII. The authors also formulate a general Prolongation Conjecture, which was already proved under appropriate non-degeneracy conditions. In Chapter VII several examples are given. First of all, the systems of first order equations for two functions of two variables are studied. There are many subcases, the detailed discussion of which is an excellent exercise for the reader. Other very interesting problems are the isometric embeddings with additional conditions. Chapter VIII collects those facts from commutative algebra and algebraic geometry that are necessary for the theory of systems of partial differential equations. From its contents we select the following subjects: a very general form of the formal Cartan- Poincaré lemma, Spencer cohomology, the graded SV-module associated to a tableau (where SV is the polynomial ring of vector space V), Koszul homology, the canonical resolution of an involutive module, the graded module associated to a higher order tableau.

The last two chapters present an introduction to the theory of overdetermined systems of partial differential equations as it has been developed over the last twenty five years. Rather than giving complete proofs, the authors prefer in general to present many examples illustrating the various methods used in the theory. A k-th order partial differential equation \(R_ k\) on a fibered manifold \(E\to X\) is a fibered submanifold of its k-th jet prolongation. Chapter IX is devoted to the basic existence theorems of Goldschmidt for \(R_ k\). A formal solution of \(R_ k\) is an element of the infinite jet prolongation of E satisfying all prolongations of the equation \(R_ k\). First the conditions which guarantee the existence of sufficiently many formal solutions are provided. They are expressed in terms of the Spencer cohomology groups of the symbol of \(R_ k\) and of its prolongations. Then for an analytic system satisfying those conditions, the convergence of formal solutions is proved and thus the existence of local solutions. The geometric examples are mostly taken from the recent papers by Goldschmidt, Gasqui and deTurck. In Chapter X, linear differential operators are studied from such a point of view. The most interesting geometric examples deal with the integrability condition of an almost CR- structure and with certain problems from Riemannian geometry.

The book under review is written in a very successful manner. The authors do not only collect all basic results from the theory and present them in a readable way, but they really intend to teach the reader how to apply them in geometry and analysis. Since the theory of exterior differential systems is a rather complicated subject, such a didactical effort is a crucial advantage of the book. It can be expected that the book will serve as the fundamental reference book in its field for many years.

Chapter I gives a review of exterior algebra and an introduction to jet bundles. An exterior differential system on a manifold M is defined as an ideal \({\mathcal I}\) in the exterior algebra of M that is closed under exterior differentiation. Usually one prescribes an independence condition for \({\mathcal I}\) in the form of a decomposable exterior p-form \(\Omega\) on M. Then the integral manifolds of (\({\mathcal I},\Omega)\) are the p-dimensional integral manifolds N of \({\mathcal I}\) satisfying \(\Omega |_ N\neq 0\). Chapter II deals with simple exterior differential systems that can be studied using ordinary differential equations. First the classical results by Frobenius, Pfaff, Darboux and others are deduced. Then the Bryant normal form of some Pfaffian systems (generated by 1-forms) is derived.

Chapter III presents the Cartan-Kähler existence theorem, which is based on repeated application of the Cauchy-Kowalewski theorem, so that all data are assumed to be real analytic. The proof follows the classical one by Kähler, but the presentation is extremely clear owing to numerous methodical improvements by the authors. In particular, the Cartan test for an ordinary integral element is clarified from several points of view. Then some geometric applications are discussed, the most important of which is the proof of the Cartan-Janet theorem on local isometric embedding of a Riemannian n-space into the Euclidean \(n(n+1)/2\)-space. The main goal of Chapter IV is to develop the formalism of linear Pfaffian systems in a form that facilitates the concrete computation in geometric problems. An arbitrary system (\({\mathcal I},\Omega)\) is said to be in involution at \(x\in M\), if there exists an ordinary integral element \(E\subset T_ xM\). The most important algebraic problems from the theory of systems in involution are reflected in the concept of a tableau. In general, a tableau is a linear subspace \(A\subset Hom(V,W)\), where V and W are vector spaces. An involutive tableau is defined purely algebraically and it is deduced that the tableau \(A_ E\) of the linearization (\({\mathcal I}_ E,\Omega_ E)\) of (\({\mathcal I},\Omega)\) at E is involutive at every ordinary integral element E. Then the linear Pfaffian systems (linear with respect to a natural affine structure) are discussed in detail. Such a system can be determined by two subbundles \(I\subset J\subset T^*M\) satisfying \(dI=0 mod\{J\}\). This setting leads directly to the concept of the torsion of (I,J). A very practical result is that a linear Pfaffian system is in involution at \(x\in M\) if and only if its torsion vanishes locally and its tableau at x is involutive. This theorem is applied to solve several geometric problems. Chapter V is devoted to the characteristic variety of an exterior differential system. Several examples illustrate its important role in geometry.

A classical conjecture by E. Cartan reads that after a finite number of prolongations every differential system becomes involutive (systems with no solutions being allowed as a special case). Precise and effective results of this type represent a complicated problem (some known results are based on certain assumptions that can be verified in the course of the prolongation procedure only). That is why the authors present a subtle definition of a prolongation sequence of an exterior differential system in Chapter VI. Then a version of the Cartan-Kuranishi prolongation theorem is proved. The best results can be deduced for a linear Pfaffian system, the tableau of which is 2-acyclic in the sense of Chapter VIII. The authors also formulate a general Prolongation Conjecture, which was already proved under appropriate non-degeneracy conditions. In Chapter VII several examples are given. First of all, the systems of first order equations for two functions of two variables are studied. There are many subcases, the detailed discussion of which is an excellent exercise for the reader. Other very interesting problems are the isometric embeddings with additional conditions. Chapter VIII collects those facts from commutative algebra and algebraic geometry that are necessary for the theory of systems of partial differential equations. From its contents we select the following subjects: a very general form of the formal Cartan- Poincaré lemma, Spencer cohomology, the graded SV-module associated to a tableau (where SV is the polynomial ring of vector space V), Koszul homology, the canonical resolution of an involutive module, the graded module associated to a higher order tableau.

The last two chapters present an introduction to the theory of overdetermined systems of partial differential equations as it has been developed over the last twenty five years. Rather than giving complete proofs, the authors prefer in general to present many examples illustrating the various methods used in the theory. A k-th order partial differential equation \(R_ k\) on a fibered manifold \(E\to X\) is a fibered submanifold of its k-th jet prolongation. Chapter IX is devoted to the basic existence theorems of Goldschmidt for \(R_ k\). A formal solution of \(R_ k\) is an element of the infinite jet prolongation of E satisfying all prolongations of the equation \(R_ k\). First the conditions which guarantee the existence of sufficiently many formal solutions are provided. They are expressed in terms of the Spencer cohomology groups of the symbol of \(R_ k\) and of its prolongations. Then for an analytic system satisfying those conditions, the convergence of formal solutions is proved and thus the existence of local solutions. The geometric examples are mostly taken from the recent papers by Goldschmidt, Gasqui and deTurck. In Chapter X, linear differential operators are studied from such a point of view. The most interesting geometric examples deal with the integrability condition of an almost CR- structure and with certain problems from Riemannian geometry.

The book under review is written in a very successful manner. The authors do not only collect all basic results from the theory and present them in a readable way, but they really intend to teach the reader how to apply them in geometry and analysis. Since the theory of exterior differential systems is a rather complicated subject, such a didactical effort is a crucial advantage of the book. It can be expected that the book will serve as the fundamental reference book in its field for many years.

Reviewer: I.Kolář (Brno)

### MSC:

58-02 | Research exposition (monographs, survey articles) pertaining to global analysis |

58A15 | Exterior differential systems (Cartan theory) |

58A17 | Pfaffian systems |

58A20 | Jets in global analysis |

58J10 | Differential complexes |

35N10 | Overdetermined systems of PDEs with variable coefficients |