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Introduction to the $$h$$-principle. (English) Zbl 1008.58001
Graduate Studies in Mathematics 48. Providence, RI: American Mathematical Society (AMS) (ISBN 0-8218-3227-1/hbk). xvii, 206 p. (2002).
In differential geometry and topology one often deals with systems of partial differential equations, as well as partial differential inequalities, which have infinitely many solutions whatever boundary conditions are imposed. In the book, homotopical methods for solving this kind of differential relations are disscused. It was discovered in the fifties that the solvability of differential relations can be often reduced to a problem of a purely homotopy-theoretic nature. One says in this case that the corresponding differential relation satisfies the $$h$$-principle. The idea of $$h$$-principle first appears in papers by Gromov and the first author in the beginning of the seventies, and the term $$h$$-principle was introduced by M. Gromov in his book “Partial differential relations”, Springer-Verlag (1986; Zbl 0651.53001). While Gromov’s book is written for experts in the theory, the reviewed book is the first broadly accessible exposition of the theory written for mathematicians who are interested in an introduction into the $$h$$-principle and its applications. The book is very readable, many motivations, examples and exercises are included. So the book is a very good text for graduate courses on geometric methods for solving partial differential equations and inequalities.
The book covers two main methods for proving the $$h$$-principle: holonomic approximation and convex integration. A special emphasis is made on applications to symplectic and contact geometry.
The book is divided into 4 parts: Part 1 (Holonomic Approximation) is an introduction into the jet spaces and holonomic approximation of their sections. Part 2 (Differential Relations and Gromov’s $$h$$-Principle) is an explanation of the $$h$$-principle in solving differential relations. As an application the Smale-Hirsch immersion theorem is proved. In Part 3 (The Homotopy Principle in Symplectic Geometry) many applications of the $$h$$-principle to symplectic and contact geometry are described. The method for proving the $$h$$-principle based on the holonomic approximation theorem works well for open manifolds. Applications to closed manifolds require an additional trick, called microextension. The holonomic approximation method also works well for differential relations which are not open, but microflexible. The most interesting applications of this types come from symplectic geometry and, for instance, the $$h$$-principle is used for Legendre, isotropic, isocontact, isosymplectic and Lagrangian immersions. Finally in Part 4 (Convex Integration) the $$h$$-principle and convex integration are studied and the Nash-Kuiper theorem is proved.

MSC:
 58Axx General theory of differentiable manifolds 58-02 Research exposition (monographs, survey articles) pertaining to global analysis 53D99 Symplectic geometry, contact geometry
Zbl 0651.53001