"Exterior differential systems" de Bryant et al. .

"Partial differential relations" de Gromov,

"Basic Ideas and concepts of differential geometry " par Alekseevskij, Vinogradov, Lychagin; in Geometry I, Springer 1991.

"Lecture notes on geometrical aspects of partial differential equations" de Zharinov, World Scientific, 1992.

Plusieurs papiers des auteurs : Chetverikov,Fliess, Lévine, Martin, Rouchon,
 
 

Voici quelques détails des ouvrages cités:
 
 
 
 

95i:53001a 53-01
Geometry. I.

Basic ideas and concepts of differential geometry.

95i:53001b 53-01
Alekseevskij, D. V.(RS-MPDI); Vinogradov, A. M.(RS-MOSC-MM); Lychagin, V. V.(RS-ALLU-CE)
Basic ideas and concepts of differential geometry.
Geometry, I, 1--264,
Encyclopaedia Math. Sci., 28,
Springer, Berlin, 1991.

References: 0 Reference Citations: 1  Review Citations: 1
 

In the book under review, the authors present the fundamental notions, ideas and methods of differential geometry. As in physics where, departing from the "big bang" hypothesis, physicists
explain the existing state of the universe, the authors of the current book make an attempt to realize a variety of modern geometric theories.

They consider differential geometry as a unified discipline which is still in the process of development and which provides a geometric interpretation of the basic objects of differential calculus on a manifold. Their survey of the basics of differential geometry is not intended for a systematic reading from the beginning to the end. This is the reason that while reading each chapter the reader can start or stop wherever it is natural for him or her. The authors start each topic with a "general discussion" and trace this discussion into precise formulas as far as possible. They consider two approaches to differential geometry: the field approach (Riemann) and the group approach (Lie and Klein) in their interaction. The synthesis of these two approaches leads to the theory of $G$-structures. The authors note the rise and evolution of basic notions of diferential geometry such as metric, connection, curvature, group of symmetries, etc. They present the theory of curves and surfaces (in both three-dimensional and multidimensional Euclidean spaces), Riemannian geometry, geometry of homogeneous spaces, geometry of differential equations, the general theory of geometric objects and $G$-structures, geometry of jets, the theory of pseudogroups, differential invariants and some other topics. They also consider some global aspects of differential geometry.

The contents of the book can be seen from its chapter titles and topics discussed in them (they are indicated in parentheses): 1. Introduction: a metamathematical view of differential geometry. 2. The geometry of surfaces (curves in Euclidean space $E\sp n$, surfaces in $E\sp 3$, and $n$-dimensional submanifolds in $E\sp {n+p}$). 3. The field approach of Riemann (intrinsic and
Riemannian geometry, manifolds and bundles, tensor fields and differentiable forms, Riemannian manifolds and manifolds with a linear connection, the geometry of symbols). 4. The group
approach of Lie and Klein. The geometry of transformation groups (symmetries in geometry, homogeneous spaces and invariant connections on them, homogeneous Riemannian and symplectic
manifolds). 5. The geometry of differential equations (geometry of a first order equation, contact geometry and Lie's theory of first order equations, the geometry of distributions, spaces of jets
and differential equations, the theory of compatibility and formal integrability, Cartan's theory of systems in involution, the geometry of infinitely prolonged equations). 6. Geometric structures
(geometric quantities and structures, principal and vector bundles and connections in them, bundles of jets). 7. The equivalence problem, differential invariants and pseudogroups (a general view on the equivalence problem, the general equivalence problem in Riemannian geometry, for geometric structures, for $G$-structures; differential invariants, pseudogroups, Lie equations, the structure of Lie pseudogroups). 8. Global aspects of differential geometry.

\{For the entire collection see MR 95i:53001a\}.                                     Reviewed by Vladislav Goldberg

92h:58007 58A15 (35N10 58-01 58-02 58A17 58G05)
Bryant, R. L.(1-DUKE); Chern, S. S.(1-MSRI); Gardner, R. B.(1-NC); Goldschmidt, H. L.(1-CLMB); Griffiths, P. A.(1-DUKE)

Exterior differential systems.

References: 0 Reference Citations: 12 Review Citations: 5
 

While the work of Elie Cartan includes some of the most significant differential geometry of this (or any) century, it is at the same time notoriously difficult to read. There are probably many
geometers who can identify with Hermann Weyl's comments that "I did not quite understand how he does this in general, though in the examples he gives the procedure is clear" and
"Nevertheless, I must admit that I found the book, like most of Cartan's papers, hard reading". The book under review is both a response to and a remedy for such difficulties. In the words of the
authors' introduction: "This book grew through our efforts to work through and appreciate Partie II of Cartan's OEuvres completes (Cartan [1953]), which we found to be full of interesting ideas
and details. Hopefully our presentation will help the study of the original work, which we cannot replace. In fact, for readers who have gone through most of this book we propose the following
problem as a final examination: Give a report on his famous five-variable paper, `Les systemes de Pfaff a cinq variables et les equations aux derivees partielles du second ordre' (Cartan,
[1910])."

Chapters I--IV of this book represent a much expanded and more thorough treatment of the topics originally discussed by Bryant, Chern and Griffiths [in Proceedings of the 1980 Beijing
Symposium on Differential Geometry and Differential Equations, Vol. 1, 2, 3 (Beijing, 1980), 219--338, Science Press, Beijing, 1982; MR 85k:58005]. An (exterior) differential system is an ideal
$I$ in the graded ring of differential forms on a smooth manifold $M$ that is closed under the operation of exterior differentiation. An integral manifold of a differential system is an (immersed)
submanifold of $M$ on which each form in the system restricts to zero. The central problem in the theory is to determine and study the integral manifolds of a given differential system.

Chapters I and II establish notation, terminology, and some basic results about differential systems in the $C\sp \infty$ category. Here the principal tool is the theorem of Frobenius, which is in
turn a consequence of the theory of ordinary differential equations. The authors define Cauchy characteristics and the corresponding completely integrable retracting system which allows one to
discover "superfluous" variables. Additional topics include a discussion of Pfaffian systems together with the associated "normal forms" of Pfaff, Darboux, Bryant, Engel, and Goursat.

In general, however, the construction of integral manifolds of a differential system cannot be accomplished by means of ordinary differential equations alone. Instead, it is usually necessary to
solve a system of partial differential equations. As a consequence, the theory of differential systems has its natural setting in the real analytic category where the classical
Cauchy-Kovalevskaya theorem provides a means of establishing existence and uniqueness. However, for the study of differential systems the Cauchy-Kovalevskaya theorem is less useful than
a coordinate-free generalization known as the Cartan-Kahler theorem. The Cartan-Kahler theorem states that for any "ordinary" integral element $E\subset T\sb xM$ of an analytic differential
system, there exists an integral manifold through $x$ whose tangent space at $x$ is $E$. In Chapter III the authors prove the Cartan-Kahler theorem and provide a version of Cartan's test for
determining when an integral element is actually ordinary. Chapter III concludes with several applications of the Cartan-Kahler theorem. In particular, the authors provide a proof of the existence
of orthogonal coordinates for any real analytic metric on a three-dimensional manifold and a proof of the Cartan-Janet isometric embedding theorem.

In many cases it is desirable that some collection of $n$ independent variables on $M$ remain independent when restricted to an integral manifold. To accomplish this, an "independence
condition" is added to the differential system by requiring that some specified decomposable $n$-form $\Omega$ not vanish on the integral manifolds. For example, consider the contact system
on the space of $k$-jets of maps from $\bold R\sp n$ to $\bold R\sp m$. Integral manifolds of the contact system on which $\Omega=dx\sp 1\wedge dx\sp 2\wedge\cdots\wedge dx\sp n$ is
nonzero then correspond to the $k$-graphs of functions from $\bold R\sp n$ to $\bold R\sp m$. A second example of a differential system with an independence condition is provided by the
canonical system on the Grassmann bundle of $n$-planes over a manifold. In fact, these examples are archetypes for a very important class of differential systems known as linear Pfaffian
systems. (These were referred to as Pfaffian systems in "good form" in the paper of Bryant, Chern and Griffiths [op. cit.; MR 85k:58305].) Linear Pfaffian systems are significant both because
they include many of the differential systems that naturally arise in practice and also because the application of Cartan's test to such systems is a relatively straightforward process. In Chapter
IV the authors study the class of linear differential systems with special emphasis on the collection of linear Pfaffian systems. There is also a discussion of the important (and subtle) notion of
differential systems in involution. A differential system $I$ with independence condition $\Omega$ is in involution at a point $x$ in $M$ if $I$ has an ordinary integral $n$-plane $E\subset T\sb
xM$ on which $\Omega$ is nonzero. When a real analytic differential system is in involution at a point $x$ in $M$, the Cartan-Kahler theorem guarantees the existence of an $n$-dimensional
integral manifold of $I$ through $x$ whose tangent space at $x$ is $E$. If the system fails to be in involution it can usually be "prolonged" to a linear Pfaffian system on the "space" of integral
$n$-planes over $M$. In Chapter IV the authors initiate a discussion of this process of prolongation under the assumption that the space of integral $n$-planes is a smooth submanifold of the
corresponding Grassmann bundle. There is then a local one-to-one correspondence between the integral manifolds of the prolonged system and the integral manifolds of the original system.
Consequently, the problem of finding the integral manifolds of the original system is "solved" if the prolonged system is in involution. If not, the prolonged system can itself be prolonged and the
new system checked for involution. After a finite number of successive prolongations one obtains a differential system that is either in involution or inconsistent. (Roughly speaking, this is the
content of the Cartan-Kuranishi prolongation theorem. The authors postpone their discussion of the general theory of prolongation until Chapter VI, Prolongation theory, where they prove a
version of the Cartan-Kuranishi theorem. In this chapter they also discuss some of the problems inherent in a general theory of prolongation and include some open questions and conjectures.)

The study of linear Pfaffian systems is closely related to the purely algebraic concept of a tableau. In Chapter IV the authors point out the connections between tableaux and linear Pfaffian
systems and define involutive tableau and the prolongation of a tableau. (The latter is analogous to the prolongation of a linear Lie algebra that occurs in the theory of $G$-structures.) Chapter IV
concludes by applying the theory that has been developed to characterize the continuous families of surfaces in $E\sp 3$ which are connected by isometries preserving the lines of curvature.

In Chapter V the authors study the properties of the real and complex characteristic varieties associated to a differential system. In the absence of Cauchy characteristics, the real characteristic
variety of a differential system consists of the collection of hyerplanes in integral $n$-planes whose extension to an integral $n$-plane is not unique. As a consequence, the characteristic variety
naturally projects onto the space of integral $n$-planes and the fibre of the projection is an algebraic variety of a projective space. If the characteristic variety of a linear Pfaffian system is empty
the system is said to be elliptic. The authors prove a local existence theorem for elliptic systems and illustrate the use of characteristic varieties in the study of triply orthogonal systems, linear
Weingarten surfaces, Darboux framings, and the isometric embedding problem. The latter is studied yet further in Chapter VII and involves recent work on isometric embeddings [Bryant, E.
Berger and Griffiths, Duke Math. J. 50 (1983), no. 3, 803--892; MR 85k:53056; Bryant, Griffiths and D. Yang, ibid. 50 (1983), no. 4, 893--994; MR 85d:53027]. The complex characteristic variety
is defined to be the collection of complex solutions to the polynomial equations that define the real characteristic variety. The authors provide some very interesting results involving the degree
and dimension of the complex characteristic variety of an involutive real analytic linear Pfaffian system. As a corollary of one of these results, they prove that if the complex characteristic variety
of a differential system is empty, then successive prolongations of the system lead either to an inconsistent system or to a Frobenius system. Chapter VII is a collection of examples that further
illustrate the general theory. These include the study of systems of first-order partial differential equations for two functions of two variables, the theory of webs, orthogonal coordinates, and the
isometric embedding problem.

Chapter VIII contains some important applications of commutative algebra and algebraic geometry to the study of linear Pfaffian systems. It is shown that the Spencer cohomology groups of a
tableau arise naturally in the study of such systems. For example, the authors prove that a tableau is involutive if and only if certain of these groups vanish. The proofs of some earlier theorems
(in particular, the author's version of the Cartan-Kuranishi prolongation theorem) are completed in Chapter VIII.

In Chapters IX and X overdetermined systems of partial differential equations are studied with the aid of Spencer cohomology. Chapter IX outlines the proof of Goldschmidt's existence theorem
for nonlinear systems of partial differential equations while Chapter X is a study of linear differential operators.

It is the reviewer's opinion that this is a very important book, one that will remain the standard reference on the subject for years to come. The authors have taken pains to help the reader
understand the intricate interplay between differential systems and partial differential equations and to place the many (and often technical) results in a historical context. Each chapter begins
with an introduction in which the authors discuss their perspective on (and evaluation of) the mathematics that is to be developed. While the role played by algebraic geometry in the theory can
be somewhat daunting to a purely "differential" geometer, the necessity of such algebraic considerations is made quite clear. In conclusion, this book should be of interest to a wide range of
readers, all the way from students who are just learning the subject, to researchers who use differential systems on a regular basis. Even those readers who do not feel up to taking the authors'
"final exam" should find their appreciation for differential systems much enhanced by this volume.

                                       Reviewed by Irl Bivens

Partial differential relations.

References: 0 Reference Citations: 15  Review Citations: 20
 

Around 1970, the world of differential geometry was astounded by the news that a young Russian by the name of Mikhael Gromov had proved that any noncompact differential manifold admits a
Riemannian metric of positive sectional curvature, and also one of negative sectional curvature. We were also told that this was achieved by a "soft" method of topological sheaves. Moreover, in one and the same setting, Gromov also proved generalizations of both the Hirsch-Smale immersion theorem and the A. Phillips submersion theorems. Many more results were promised. Slowly, Gromov's papers (some in collaboration with Ya. M. Eliashberg and V. A. Rokhlin) filtered to the West in the early seventies. Here are a sample of those particularly relevant to the present review: the author [in Actes du Congres International des Mathematiciens, Tome II (Nice, 1970), 221--225, Gauthier-Villars, Paris, 1971; MR 54 #8709; Izv. Akad. Nauk SSSR Ser. Mat. 33 (1969), 707--734; MR 41 #7708], the author and Eliashberg [Math. USSR-Izv. 5 (1971), 615--639; MR 46 #903], the author [ibid. 7 (1973), no. 2, 329--343; MR 54 #1323] and the author and Rokhlin [Russian Math. Surveys 25 (1970), no. 5, 1--57; MR 44 #7571]. After a lapse of some fifteen years, the author has now presented what would appear to be his valedictory statement on the subject. Within the covers of the volume under review, he has deepened, generalized and synthesized the materials from the diverse earlier publications to arrive at a coherent account starting from first principles. The appearance of this book is a major event in geometry during the past decade.

The aim and scope of the book are succinctly set forth in the foreword: "The classical theory of partial differential equations is rooted in physics, where equations (are assumed to) describe the
laws of nature. Law-abiding functions, which satisfy such an equation, are very rare in the space of all admissible functions.$\,\ldots$Moreover, some additional conditions often insure the
uniqueness of solutions.$\,\ldots$We deal in this book with a completely different class of partial differential equations (and more general relations) which arise in differential geometry rather
than in physics. Our equations are, for the most part, under-determined (or, at least, behave like those) and their solutions are rather dense in spaces of functions. We solve and classify
solutions of these equations by means of direct (and not so direct) geometric constructions. Our exposition is elementary and the proofs of the basic results are self-contained." The partial
differential relations alluded to above are usually either equations or inequalities. A typical example of the former is the system of partial differential equations arising from the isometric
imbedding problem for Riemannian manifolds. Let $M$ be an $n$-dimensional Riemannian manifold with metric $g=\sum\sp n\sb {i,j=1}g\sb {ij}dx\sp idx\sp j$. Let $f\colon M\to\bold R\sp q$ be a $C\sp \infty$ map into a high-dimensional Euclidean space, and let $f=(f\sp 1,\cdots,f\sp q)$. We want $f$ to be injective and that its components $\{f\sp a\}$ satisfy: $(*)$ $\sum\sp q\sb
{a=1}(\partial f\sp a/\partial x\sp i)(\partial f\sp a/\partial x\sp j)=g\sb {ij}$ for all $i,j=1,\cdots,n$. This system $(*)$ expresses of course the fact that the induced metric on $f(M)$ equals $g$.
Here we have $\frac 12n(n+1)$ equations in the $q$ unknowns $\{f\sp 1,\cdots,f\sp q\}$. Since $q$ will be taken to be large, $(*)$ is grossly under-determined. A typical example of the kind of
partial differential inequalities treated in this book is the following: Given differential manifolds $V$, $W$ with dimensions $n$ and $q$, respectively $(n\leq q)$, we ask if there is an immersion
$f\colon V\to W$. In terms of local coordinates, $df$ can be represented as an $n\times q$ matrix. Let $\{D\sb i\}$ be the ${q\choose n}$ submatrices of dimension $n\times n$ in $df$. Then the
property of $f$'s being an immersion is expressed by the following inequality to be satisfied at each point: $(\#)$ $\sum\sb i(\det D\sb i)\sp 2>0$.

Technically, the formulation of these problems takes a different form. In Part 1 of the book, which comprises four sections, one finds a general discussion of this formalism together with a survey of the basic problems and results. Thus let $p\colon X\to V$ be a smooth fibration and let $X\sp {(r)}$ be the space of germs of $r$-jets of smooth sections $V\to X$. Thus each $X\sp {(r)}$ is a bundle over $X$ whose fibre at each $x\in X$ consists of all linear maps $\psi$ from $T\sp r\sb {p(x)}V$ (the tangent space at $p(x)$ of $V$ of order $r$) to $T\sp r\sb xX$, such that for all $1\leq s\leq r$, $\psi(T\sp s\sb {p(x)}V)\subset T\sp s\sb xX$. By taking the $r$th order jet of a smooth section $f\colon V\to X$, we get a smooth section $J\sp rf\colon V\to X\sp {(r)}$. The set of all such sections $\{J\sp rf\}$ as $f$ varies over all sections $f\colon V\to X$ is called the set of holonomic sections. Clearly $J\sp rf$ is locally nothing but the string of all partial derivatives of $f$ up to order $r$. A differential relation on the sections of $p\colon X\to V$ is just a subset $\scr R\subset X\sp {(r)}$, and a solution of $\scr R$ is by definition a section $f\colon V\to X$ such that $J\sp rf(V)\subset\scr R$. Thus we may identify the solutions of a differential relation $\scr R$ with the holonomic sections $V\to X\sp {(r)}$ which map into $\scr R$. Usually it is easy to construct a continuous section $f\colon V\to\scr R$, or else one such is given. Then the obvious way to obtain a solution of $\scr R$ is to deform by homotopy the section $f$ into a holonomic one (abbreviation: homotop $f$ into a holonomic section), if this is possible. We say $\scr R$ satisfies the homotopy principle ($h$-principle for short) if every continuous section $V\to\scr R$ can be homotoped into a holonomic section. The main goal of this book is to show, often surprisingly, that in a wide variety of situations, the $h$-principle holds for the partial differential relation at hand.
For the sake of simplicity, we take up three examples to give a flavor of this work. (I) Let $p\colon X\to V$ be a holomorphic fibre bundle with a Stein manifold $V$ as base, with a complex Lie
group $G$ as structure group, and with $G/H$ as fibre where $H$ is a complex Lie subgroup of $G$. Define the relation $\scr R\subset X\sp {(1)}$ to consist of complex linear maps from the
tangent space $T\sb {p(x)}V$ to $T\sb xX$ for each $x\in X$. The Cauchy-Riemann equations imply that every holonomic section of $\scr R$ must be holomorphic. The celebrated Grauert-Oka principle asserts that the $h$-principle holds for this $\scr R$ [H. Grauert, Math. Ann. 133 (1957), 450--472; MR 20 #4660]. (II) Let $V$, $W$ be differential manifolds of dimensions $n$ and $q$, respectively, $n\leq q$. Let $X=V\times W$ and let $p\colon X\to V$ be the obvious projection. Now a section of $X\sp {(1)}\to V$ is a map $v\to((v,w),\psi)$, where $v\in V$, $w\in W$ and $\psi$ is a linear map $T\sb vV\to T\sb wW$. A holonomic section is then a map $v\to((v,f(v))$, $df\colon T\sb vV\to T\sb {f(v)}W)$, where $f$ is a map from $V$ to $W$, and hence the holonomic section may be simply identified with the pair $(f,df)$. Now define the immersion relation $\scr R$ on this $X\sp {(1)}$ to consist of only those $((v,w),\psi)$ where $\psi$ is injective.
It follows that the holonomic sections of $\bold R$ in this case consist of immersions $V\to W$. The immersion theory of Hirsch and Smale [M. W. Hirsch, Trans. Amer. Math. Soc. 93 (1959),
242--276; MR 22 #9980] guarantees that the $h$-principle is valid if either $n<q$ or if $V$ is open. (III) Let $V$ be a Riemannian manifold of dimension $n$, and let $X=V\times\bold R\sp q$,
where $q\geq\frac 12(n+2)(n+3)$. As usual, $p\colon X\to V$ is given by the obvious projection. Now define $\scr R\subset X\sp {(2)}$ to be the set of all $((v,y),\psi)$ where $v\in V$,
$y\in\bold R\sp q$ and $\psi$ is a nonsingular linear map $T\sp 2\sb vV\to T\sp 2\sb y\bold R\sp q$ such that the restriction of $\psi$ to $T\sb vV$ is an isometric linear map into $T\sb y\bold R\sp q$. With the same reasoning as in (II), we see immediately that a holonomic section of this $\scr R$ is just an isometric immersion of $V$ into $\bold R\sp q$ whose second order differential is everywhere nonsingular. Such immersions are called free isometric immersions, and they originated with the classical work of J. F. Nash, Jr. [Ann. of Math. (2) 63 (1956), 20--63; MR 17, 782].
Now a theorem of the author states that this $\scr R$ also obeys the $h$-principle.

Of course the book considers a wide range of topics (submersions, $C\sp \infty$ foliations, isometric imbeddings of Riemannian manifolds, contact structures, symplectic structures, etc.), but
through these three examples one can already perceive many of the central features of this work. First, it does not give a proof of (I), i.e., the Grauert-Oka principle. This is perhaps due to the
fact that this principle is a theorem in the theory of over-determined systems. But by the same token, the author does infuse every topic that gets discussed in the book with new results or a new viewpoint. Second, while the whole book is concerned with only one problem (when does the $h$-principle hold?), the consideration of the $h$-principle provides only the philosophical backbone to the book. Technically, each topic is treated from a perspective (sometimes up to three different perspectives, as in the case of the Hirsch-Smale theorem in (II)) uniquely its own. For example,
(III) requires careful attention to the hard analysis inherent in the Nash imbedding theorem, while (II) is of course free from such considerations. Third, while some of the results treated in this
book are partly or wholly known ((III) and (II) respectively, for example), the new ideas and improvements the author brings to these well-known topics are considerable. Thus he not only gives the Hirsch-Smale theorem three different proofs (recall Atiyah's dictum: if you only have one proof for a theorem then you cannot say you understand it very well), but the deeper understanding so achieved immediately allows him not only to treat submersion, $k$-mersions, etc. in the same setting, but also to draw new conclusions as well; for example, the theorem of the author and Eliashberg that holomorphic immersions of Stein manifolds into complex Euclidean space of strictly higher dimension also obey the $h$-principle. As another example, the result in (III) above on free isometric immersions leads immediately to the smallest known dimension of the receiving Euclidean space for isometric immersions of Riemannian manifolds. Last but not least, the reader will be happily surprised at every turn that, running through the many seemingly unrelated topics that show up in this book, there is the common thread of the $h$-principle (and sometimes more).
The ability to draw together disparate topics under one roof is in fact a prominent feature of the author's work.

The heart of the book is Part 2, also comprising four sections. Its concern is the methods to prove the $h$-principle, to wit, removal of singularities, continuous sheaves, inversion of differential
operators, and convex integration. These are, appropriately enough, highly technical matters so that a short discussion of these would make no sense and a sufficiently detailed discussion would be impossibly long. Perhaps a few peripheral remarks would suffice. The author's aim here is of course to prove theorems that would guarantee the validity of the $h$-principle in the most general and in the maximum number of situations. These theorems therefore tend to be abstract and do not make easy reading. The reviewer feels that reading the first four of the papers cited at the beginning could be helpful. In these papers, the intended applications of each method are always stated clearly, and the method comes through in a more transparent fashion because the machinery is less sophisticated. True, the exposition in these papers is sometimes sketchy in the technical details, but for the purpose of acquiring a general idea of the arguments this could even be an advantage. For example, to see how the Hirsch-Smale theorem could be proved by the method of continuous sheaves or convex integration, reading the 1969 paper [op. cit.; MR 41 #7708] or the 1973 paper [op. cit.; MR 54 #1323] would seem to yield this information much more readily. Another relevant remark is that Part 3 of the book contains the most substantive applications of the method of inversion of differential operators (and therefore to a certain extent, of the method of continuous sheaves), so perhaps they should be read simultaneously.

Finally, a few words about Part 3 on isometric $C\sp \infty$-immersions. This part occupies 140 out of the 360 pages of the book, and this fact makes this topic unique among the many
applications of the general theory developed here. Note that the explicit mention of "$C\sp \infty$" is significant: this is to distinguish it from the $C\sp 1$-immersion theory of Nash-Kuiper. The latter is of a totally different character; for example, every $n$-dimensional Riemannian manifold can be $C\sp 1$ isometrically immersed into $\bold R\sp {2n}$. In this book, the $C\sp 1$ theory is taken up in the context of convex integration in Part 2 (this part of the author's work seems to be appearing in print for the first time). What show up in Part 3 are various refinements of, and additions to the pioneering work of Nash concerning isometric imbedding of Riemannian manifolds into Euclidean space: improvements on the receiving dimension (alluded to above), what happens in low dimensions, the role of the second fundamental form, the case of indefinite metrics, and the case of a symplectic form (replacing the Riemannian metric). For this part, the paper of Gromov-Rokhlin referred to at the end of the first paragraph can serve as a good introduction.

As we mentioned above, the author intended this book to be an elementary exposition. This should not be taken literally. One should rather approach this as a research monograph where new
ideas turn up almost in every page. Many of these ideas will undoubtedly inspire further developments.

                                      Reviewed by Hung-Hsi Wu

93h:58171 58G37 (35A30 35Qxx 58F07 58H05)
Zharinov, V. V.(RS-AOS)

Lecture notes on geometrical aspects of partial differential equations.

References: 0  Reference Citations: 1  Review Citations: 0
 

The author treats PDE as a differential manifold $(M,CTM)$, where $M$ is a smooth finite- or infinite-dimensional manifold and $CTM$ is a finite-dimensional involutive distribution on $M$
(Cartan distribution). Morphisms of the category of differential manifolds are Lie-Backlund mappings. Symmetries of PDE are described in terms of Lie-Backlund automorphisms. The standard technique of homological algebra yields the $C$-spectral sequence associated with $M$ and Cartan forms on open subsets of $M$ [A. M. Vinogradov, Dokl. Akad. Nauk SSSR 238 (1978), no. 5, 1028--1031; MR 81d:58060]. Conservation laws, functionals, a characteristic mapping and the Euler-Lagrange operator are elements of the $C$-spectral sequence. The proper generalization of the Green theorem for the Laplace equation yields the relation between conservation laws and infinitesimal symmetries and plays the same role as the Noether theorem does in the case of a Lagrangian system. Backlund transformations are discussed from the point of view of reductions of the direct product of differential manifolds. Examples range from heat and Burgers equations to the Nambu classical string model and include the Lagrangian formalism and Hamiltonian equations.

The author's purpose is to give a short simple introduction to a domain of infinite-dimensional differential geometry, providing a natural framework for the study of specific properties of PDE and explaining them in clear, precise language. Even if that approach does not help much in actual calculations, it gives a better understanding of the whole picture and presents these calculations in a neat, unified way.

                                     Reviewed by Vladimir Mikhalev

Index pour ce cours de dea:  http://picard.ups-tlse.fr/~roche/enseignement/dea2003/index.html