Introduction to the h-Principle

The Gromov “h-principle” was introduced in the 80’s, in Gromov’s book on partial differential relations. This is really a principle, not a theorem, and it should be adapted to every new application. It describes the behaviour of a system of partial differential equations or inequalities having a big space of solutions (e.g., dense in the space of functions or fields considered). It was inspired by work on the C1-isometric embedding theory (Nash and Kuiper) and the immersion theory in differential topology (Smale and Hirsch). The Gromov book is not easy to understand. The book under review offers the reader a nice and understandable treatment of two methods based on the “h-principle” – the holonomic approximation methods and convex integration theory (which was treated in a more general situation in the book by D. Spring, published in 1998). These two methods cover many interesting applications. The book contains a discussion of applications of the holonomic approximation method in symplectic and contact geometry, including a review of basic notions needed for the applications. The book is very interesting, readable and should be of interest to more than just geometers.

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