By Y. Eliashberg

In differential geometry and topology one frequently bargains with structures of partial differential equations, in addition to partial differential inequalities, that experience infinitely many options no matter what boundary stipulations are imposed. It used to be chanced on within the fifties that the solvability of differential family (i.e. equations and inequalities) of this sort can frequently be decreased to an issue of a basically homotopy-theoretic nature. One says hence that the corresponding differential relation satisfies the $h$-principle. recognized examples of the $h$-principle, the Nash-Kuiper $C^1$-isometric embedding idea in Riemannian geometry and the Smale-Hirsch immersion thought in differential topology, have been later remodeled through Gromov into strong common tools for setting up the $h$-principle.The authors conceal major tools for proving the $h$-principle: holonomic approximation and convex integration. The reader will locate that, with a couple of remarkable exceptions, so much cases of the $h$-principle should be taken care of by way of the equipment thought of the following. a different emphasis within the publication is made on functions to symplectic and make contact with geometry. Gromov's recognized publication ""Partial Differential Relations"", that is dedicated to an analogous topic, is an encyclopedia of the $h$-principle, written for specialists, whereas the current ebook is the 1st greatly available exposition of the speculation and its purposes. The booklet will be a good textual content for a graduate path on geometric equipment for fixing partial differential equations and inequalities. Geometers, topologists and analysts also will locate a lot worth during this very readable exposition of a tremendous and memorable subject

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**Extra resources for Introduction to the h-principle**

**Example text**

0 0 0 = 0 · ∂F ∂ y¯ n+1 , ∂F ∂ y¯ and hence it does not vanish near the ﬁber Xrr+1 (z). 2 can be found in [Gr86]. Chapter 3 Holonomic Approximation The Holonomic Approximation Theorem which we discuss in this chapter shows that in some sense there are unexpectedly many holonomic sections near any submanifold A ⊂ V of positive codimension. 1. Main theorem Question: Is it possible to approximate any section F : V → X (r) by a holonomic section? In other words, given an r-jet section and an arbitrarily small neighborhood of the image of this section in the jet space, can one ﬁnd a holonomic section in this neighborhood?

Proof. Let us ﬁx the notation. Denote by Grm,n W the manifold of all (m, n)-ﬂags on W , where each ﬂag is a pair of tangent planes (Lm , Ln ) in Tw W such that Ln ⊂ Lm . Denote by π and π the projections Grm,n W → Grm W and Grn,m W → Grn W . Set A¯ = {(L, L) | L ∈ A ⊂ Grm W, L ∈ Grn L } ⊂ Grn,m W, where A is the set implied by the deﬁnition of m-completeness. Note that ¯ = A and π(A) ¯ = A. π(A) Let Gt : V → Grn W be the homotopy between the tangential lift G0 = Gdf0 of the embedding f0 and the map G1 : V → A.

1. (Smale’s sphere eversion, [Sm58]) The map r ◦ inv ◦ iV : V → R3 , which inverts V outside in, is regularly homotopic to the inclusion iV : V → R3 . 2. Smale’s sphere eversion 39 Remarks 1. This counter-intuitive statement is a corollary of S. Smale’s celebrated theorem [Sm58]. e. via a family of smooth, but possibly self-intersecting surfaces. One can follow the proof below to actually construct this eversion. However, there are much more eﬃcient ways to do that. The explicit process of the eversion became the subject of numerous publications, videos and computer programs.