Convex Integration Theory: Solutions to the h-principle in geometry and topologyDavid Spring Birkhäuser, 6 dec 2012 - 213 pagina's §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. |
Inhoudsopgave
1 | |
Convex Hulls | 19 |
Analytic Theory | 33 |
Open Ample Relations in Spaces of 1Jets | 49 |
Microfibrations 71 | 70 |
The Geometry of Jet spaces | 87 |
Convex Hull Extensions | 101 |
Ample Relations | 121 |
Systems of Partial Differential Equations | 165 |
Relaxation Theorem 201 | 200 |
207 | |
Overige edities - Alles bekijken
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 Gedeeltelijke weergave - 2010 |
Convex Integration Theory: Solutions to the H-principle in Geometry and Topology David Spring Fragmentweergave - 1998 |
Veelvoorkomende woorden en zinsdelen
a₁ affine bundle ample relations Applying associated base manifold C-structure Chapter charts codimension codimension 1 tangent compact Complement constant homotopy construction continuous function continuous lift continuous map Conv convex hull extensions Convex Integration theory coordinates Corollary covering homotopy deformation retract denote differential inclusion F₁ fiber following properties obtain formal solution Furthermore Gromov 18 h-principle h-stability theorem H₁ hence holonomic section homotopy F homotopy of formal in-path induces integral representation jet spaces Lemma microfibration neighbourhood notation Op₁K open set parametrized principal subspace product bundle projection map proof of Theorem Proposition prove r-jets Rª-bundle relative theorem rth order section ƒ Serre fibration short maps small homotopy smooth manifold strictly short strictly surrounds sufficiently small Suppose surrounding paths systems of PDEs tangent hyperplane field topology vector W₁ weak homotopy equivalence