Author	Spring, David. author
Title	Convex Integration Theory [electronic resource] : Solutions to the h-principle in geometry and topology / by David Spring
Imprint	Basel : Birkhรคuser Basel : Imprint: Birkhรคuser, 1998
Connect to	http://dx.doi.org/10.1007/978-3-0348-8940-7
Descript	VIII, 213 p. 2 illus. online resource

ยง1. 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 succesยญ sive methods subsumed the previous methods. Each method has its own distinct foundation, based on an independent geometrical or analytical insight. Conseยญ 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 parยญ 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

1 Introduction -- ยง1 Historical Remarks -- ยง2 Background Material -- ยง3 h-Principles -- ยง4 The Approximation Problem -- 2 Convex Hulls -- ยง1 Contractible Spaces of Surrounding Loops -- ยง2 C-Structures for Relations in Affine Bundles -- ยง3 The Integral Representation Theorem -- 3 Analytic Theory -- ยง1 The One-Dimensional Theorem -- ยง2 The C?-Approximation Theorem -- 4 Open Ample Relations in Spaces of 1-Jets -- ยง1 Cยฐ-Dense h-Principle -- ยง2 Examples -- 5 Microfibrations -- ยง1 Introduction -- ยง2 C-Structures for Relations over Affine Bundles -- ยง3 The C?-Approximation Theorem -- 6 The Geometry of Jet spaces -- ยง1 The Manifold X? -- ยง2 Principal Decompositions in Jet Spaces -- 7 Convex Hull Extensions -- ยง1 The Microfibration Property -- ยง2 The h-Stability Theorem -- 8 Ample Relations -- ยง1 Short Sections -- ยง2 h-Principle for Ample Relations -- ยง3 Examples -- ยง4 Relative h-Principles -- 9 Systems of Partial Differential Equations -- ยง1 Underdetermined Systems -- ยง2 Triangular Systems -- ยง3 C1-Isometric Immersions -- 10 Relaxation Theorem -- ยง1 Filippovโ{128}{153}s Relaxation Theorem -- ยง2 C?-Relaxation Theorem -- References -- Index of Notation