Convex Integration Theory: Solutions to the H-principle in Geometry and TopologyBirkhäuser Verlag, 1998 - 212 pagina's This book provides a comprehensive study of convex integration theory in immersion-theoretic topology. Convex integration theory, developed originally by M. Gromov, provides general topological methods for solving the h-principle for a wide variety of problems in differential geometry and topology, with applications also to PDE theory and to optimal control theory. Though topological in nature, the theory is based on a precise analytical approximation result for higher order derivatives of functions, proved by M. Gromov. This book is the first to present an exacting record and exposition of all of the basic concepts and technical results of convex integration theory in higher order jet spaces, including the theory of iterated convex hull extensions and the theory of relative h-principles. The book should prove useful to graduate students and to researchers in topology, PDE theory and optimal control theory who wish to understand the h-principle and how it can be applied to solve problems in their respective disciplines. |
Inhoudsopgave
Introduction | 1 |
Convex Hulls | 19 |
Analytic Theory | 33 |
Copyright | |
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Convex Integration Theory: Solutions to the H-Principle in Geometry and Topology David Spring Gedeeltelijke weergave - 1998 |
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Veelvoorkomende woorden en zinsdelen
a₁ affine bundle ample relations 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 Gromov 18 h-principle h-stability theorem H₁ hence holonomic section homotopy F homotopy of formal induces integral representation jet spaces Lemma Let f microfibration neighbourhood notation Op₁K open set parametrized principal subspace product bundle projection map proof of Theorem properties are satisfied Proposition prove R-bundle r-jets Relaxation Theorem respect to f rth order section g 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