Convex integration theory : solutions to the h-principle in geometry and topology
著者
書誌事項
Convex integration theory : solutions to the h-principle in geometry and topology
（Modern Birkhäuser classics）
Birkhäuser, 2010
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注記
"Reprint of the 1998 edition"
"Originally published under the same title as volume 92 in the Monographs in mathematics series"--T.p. verso
Includes bibliographical references (p. 207-209) and index
内容説明・目次
内容説明
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.
目次
1 Introduction 2 Convex Hulls 3 Analytic Theory 4 Open Ample Relations in Spaces of 1-Jets5 Microfibrations 6 The Geometry of Jet spaces7 Convex Hull Extensions 8 Ample Relations 9 Systems of Partial Differential Equations10 Relaxation Theorem References Index Index of Notation
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