To André Haefliger with admiration Holonomic approximation and Gromov’s h-principle

In 1969 M. Gromov in his PhD thesis [G1] greatly generalized Smale-Hirsch-Phillips immersionsubmersion theory (see [Sm],[Hi], [Ph]) by proving what is now called the h-principle for invariant open differential relations over open manifolds. Gromov extracted the original geometric idea of Smale and put it to work in the maximal possible generality. Gromov’s thesis was brought to the West by A. Phillips and was popularized in his talks. However, most western mathematicians first learned about Gromov’s theory from A Haefliger’s article [H]. The current paper is devoted to the same subject as the papers of Gromov and Haefliger. It seems to us that we further purified Smale-Gromov’s original idea by extracting from it a simple but very general theorem about holonomic approximation of sections of jet-bundles (see Theorem 1.2.1 below). We show below that Gromov’s theorem as well as some other results in the h-principle spirit are immediate corollaries of Theorem 1.2.1. 1 1 Holonomic approximation 1.1 Jets and holonomy Given a C-smooth fibration p : X → V , we denote by X(r) the space of r-jets of smooth sections f : V → X and by J f : V → X (r) the r-jet of a section f : V → X. When the fibration X = V × W → V is trivial then the space X(r) is sometimes denoted by J(V,W ), and called the space of r-jets of maps V → W . A section F : V → X(r) is called holonomic if it has the form J f for a section f : V → X. The correspondence f 7→ J r f defines the derivation map The authors are partially supported by the National Science Foundation