Convex Integration Theory: Solutions to the H-Principle in Geometry and TopologySpringer Science & Business Media, 1998 - 212 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. |
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 - 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 base manifold C-structure Chapter charts codimension codimension 1 tangent compact Complement constant homotopy construction continuous lift continuous map Conv convex hull extensions Convex Integration theory Convm coordinates Corollary covering homotopy deformation retract denote differential inclusion F₁ fiber following properties obtain formal solution Furthermore geometrical h-principle h-stability theorem H₁ hence holonomic section homotopy F homotopy of formal induces IntConv integral representation jet spaces Lemma Let f microfibration neighbourhood notation Op₁K open set parametrized principal subspace product bundle projection map proof of Theorem Proposition prove R-bundle r-jets relation RC relative Relaxation Theorem respect to f rth order Serre fibration short maps small homotopy smooth manifold strictly short strictly surrounds sufficiently small Suppose surrounding paths systems of PDEs tangent hyperplane field topology weak homotopy equivalence
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