# Convex Integration Theory

## Solutions to the h-principle in geometry and topology

• David Spring
Book

Part of the Monographs in Mathematics book series (MMA, volume 92)

1. Front Matter
Pages i-viii
2. David Spring
Pages 1-18
3. David Spring
Pages 19-32
4. David Spring
Pages 33-48
5. David Spring
Pages 49-69
6. David Spring
Pages 71-86
7. David Spring
Pages 87-99
8. David Spring
Pages 101-120
9. David Spring
Pages 121-164
10. David Spring
Pages 165-199
11. David Spring
Pages 201-206
12. Back Matter
Pages 207-213

### Introduction

§1. Historical Remarks Convex Integration theory, first introduced by M. Gromov [17], is one of three general methods in immersion-theoretic topology for solving a broad range of problems in geometry and topology. The other methods are: (i) Removal of Singularities, introduced by M. Gromov and Y. Eliashberg [8]; (ii) the covering homotopy method which, following M. Gromov's thesis [16], is also referred to as the method of sheaves. The covering homotopy method is due originally to S. Smale [36] who proved a crucial covering homotopy result in order to solve the classification problem for immersions of spheres in Euclidean space. These general methods are not linearly related in the sense that succes­ sive methods subsumed the previous methods. Each method has its own distinct foundation, based on an independent geometrical or analytical insight. Conse­ quently, each method has a range of applications to problems in topology that are best suited to its particular insight. For example, a distinguishing feature of Convex Integration theory is that it applies to solve closed relations in jet spaces, including certain general classes of underdetermined non-linear systems of par­ tial differential equations. As a case of interest, the Nash-Kuiper Cl-isometrie immersion theorem ean be reformulated and proved using Convex Integration theory (cf. Gromov [18]). No such results on closed relations in jet spaees can be proved by means of the other two methods.

### Keywords

Differential topology Manifold Topology differential geometry equation function geometry theorem

• David Spring
• 1