This important work is a far-reaching development of ideas and results in differential geometry and topology due to Whitney, Nash, Kuiper, Eliashberg, Smale, Hirsch, Thom, Weinstein, the author and many others. It deals with many questions about immersions, in particular of isometric \(C^{\infty}\)-immersions of Riemannian manifolds. The discussion is squarely based upon the usual description of manifolds; the language of jets, fiber bundles, homotopy, prolongation, genericity; the systems of partial differential equations (and inequalities) of geometry; various function space topologies (or quasi-topologies) and approximation therein, infinite dimensional implicit function theorem; singularities and stratification;...
The basic notion of a differential relation is as follows. Let \(p: X\to V\) be a \(C^{\infty}\)-fibering for manifolds \(V\) and \(X\); let \(X^{(r)}\) be the space of \(r\)-jets of germs of smooth sections \(f: V\to X\). A section \(\phi : V\to X^{(r)}\) is called holonomic if it is the \(r\)-jet of some \(C^{(r)}\)-section \(f: V\to X\). Thus the space \(X^{(1)}\) of 1-jets consists of linear maps \(L: T_ v(V)\to T_ x(X)\), \(v=p(x)\in V\), of tangent spaces, such that \(D_ p\circ L=\) identity map of \(T_ v(V)\) and \(D_ p\) is the differential of the map \(p\). A differential relation imposed on sections \(f: V\to X\) is a subset \(R\subset X^{(r)}\); \(r\) is the order of \(R\). A \(C^{(r)}\)-section \(f\) is said to be a solution of \(R\) if the \(r\)-jet of \(f\), \(Jr_ f: V\to X^{(r)}\), maps \(V\) into \(R\). Thus solutions of \(R\) are naturally identified with holonomic sections \(V\to R\). \(R\) is said to satisfy the homotopy principle (\(h\)-principle) if every continuous section \(V\to R\) is homotopic to a holonomic section \(V\to R\) by a continuous homotopy of sections \(V\to R\). For complex-analytic manifolds, there is the Cauchy-Riemann relation \(R\subset X^{(1)}\) which consists of the complex-linear maps \(T_ v(V)\to T_ x(X)\) in \(X^{(1)}\). The \(h\)-principle then reduces to Oka’s principle. A chapter is devoted to holomorphic immersions of Stein manifolds.
The book is organised in an original way.
Part I is “A survey of basic problems and results”. It contains a precise formulation of the main theorems and results to be proven later as well as many remarks, examples, “exercises” and corollaries that describe the history of the subject, propose open questions and introduce some of the notions and techniques that come into play later on.
Part II consists of “Methods to prove the \(h\)-principle”. This contains a wealth of new ideas and concepts, some of which are very abstract. The chapters are as follows: 1. Removal of singularities, dealing with immersions, submersions, foldings, holomorphic immersions of Stein manifolds. 2. Continuous sheaves and the notions of flexibility, microextensions, equivariance. 3. Inversion of differential operators, including far-reaching developments of work of Nash and the algebraic solution of differential equations. 4. Convex integration.
Part III is entitled “Isometric \(C^{\infty}\)- immersions” and contains a great many results spanning a variety of different questions, all too often in the form of “exercises”.
Clearly, a great deal of effort has gone into the writing of this book, but it is nevertheless very difficult to read. There are several reasons for this. The book is full of new ideas and techniques, some quite abstract, which are applied to very difficult concrete questions such as the determination or estimation of the best possible value of dimensional parameters. Many proofs, especially in Part III, are left to the reader as “exercises”, and working these out actually requires an unusual amount of effort and detailed knowledge of many things. The whole work is an extraordinary accomplishment that should influence geometry and analysis for a long time.

Reviewer: E. J. Akutowicz (Berlin)