Convex Integration Theory: Solutions to the h-principle in geometry and topologySpringer Science & Business Media, 2 dec 2010 - 213 pagina's §1. Historical Remarks Convex Integration theory, ?rst 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 classi?cation problem for immersions of spheres in Euclidean space. These general methods are not linearly related in the sense that succ- sive methods subsumed the previous methods. Each method has its own distinct foundation, based on an independent geometrical or analytical insight. Con- 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 ConvexIntegrationtheoryisthatitappliestosolveclosed relationsinjetspaces, including certain general classes of underdetermined non-linear systems of par- 1 tial di?erential equations. As a case of interest, the Nash-Kuiper C -isometric immersion theorem can be reformulated and proved using Convex Integration theory (cf. Gromov [18]). No such results on closed relations in jet spaces can be proved by means of the other two methods. On the other hand, many classical results in immersion-theoretic topology, such as the classi?cation of immersions, are provable by all three methods. |
Inhoudsopgave
1 | |
CHAPTER 2 CONVEX HULLS | 19 |
CHAPTER 3 ANALYTIC THEORY | 33 |
CHAPTER 4 OPEN AMPLE RELATIONS IN 1JET SPACES | 49 |
CHAPTER 5 MICROFIBRATIONS | 70 |
CHAPTER 6 THE GEOMETRY OF JET SPACES | 87 |
CHAPTER 7 CONVEX HULL EXTENSIONS | 101 |
CHAPTER 8 AMPLE RELATIONS | 121 |
CHAPTER 9 SYSTEMS OF PARTIAL DIFFERENTIAL EQUATIONS | 165 |
CHAPTER 10 RELAXATION THEORY | 200 |
References | 207 |
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Overige edities - Alles bekijken
Convex Integration Theory: Solutions to the h-principle in geometry and topology David Spring Gedeeltelijke weergave - 2012 |
Convex Integration Theory: Solutions to the H-Principle in Geometry and Topology David Spring Gedeeltelijke weergave - 1998 |
Convex Integration Theory: Solutions to the H-principle in Geometry and Topology David Spring Fragmentweergave - 1998 |
Veelvoorkomende woorden en zinsdelen
addition affine Applying approximation associated assume base bundle C-structure Chapter charts closed codimension compact Complement component conclusion condition connected Consequently construction continuous map convex hull extensions coordinates Corollary corresponding covering defined denote derivatives differential Employing equivalence example existence fiber following properties follows formal solution function Furthermore G F(R g j g m G Rq G T(R geometry Gromov h-principle hence holonomic section homotopy hyperplane field immersion inclusion induces Lemma lies locally manifold microfibration neighbourhood notation Note obtain pair parametrized particular path principal subspace problem projection map proof properties Proposition prove r-jets relation relative theorem Remark respect restriction result Rq-bundle satisfies sequence smooth solves space strictly short strictly surrounds sufficiently small Suppose tangent Theorem topology vector