Selections Reprinted from Mathematical

MR0864505 (90a:58201) 58G99; 35A99, 35B99, 53C42, 58-02 Gromov, Mikhael Partial differential relations. (English) Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], 9. Springer-Verlag , Berlin, 1986. x+363 pp. $60.00. ISBN 3-540-12177-3 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. Èliashberg 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 Congrès International des Mathématiciens, Tome II (Nice, 1970), 221–225, Gauthier-Villars, Paris, 1971;MR0420697 (54 #8709); Izv. Akad. Nauk SSSR Ser. Mat. 33 (1969), 707–734;MR0263103 (41 #7708)], the author and Èliashberg [Math. USSR-Izv. 5 (1971), 615–639;MR0301748 (46 #903)], the author [ibid. 7 (1973), no. 2, 329– 343;MR0413206 (54 #1323)] and the author and Rokhlin [Russian Math. Surveys 25 (1970), no. 5, 1–57;MR0290390 (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. . . .Moreover, some additional conditions often insure the uniqueness of solutions. . . . 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