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Convex Integration Theory : Solutions to the h-principle in geometry and topology / edited by David Spring
(Monographs in Mathematics. ISSN:22964886 ; 92)

版	1st ed. 1998.
出版者	Basel : Birkhäuser Basel : Imprint: Birkhäuser
出版年	1998
本文言語	英語
大きさ	VIII, 213 p. 2 illus : online resource
冊子体	Convex integration theory : solutions to the h-principle in geometry and topology / David Spring
著者標目	Spring, David editor SpringerLink (Online service)
件　名	LCSH:Topology FREE:Topology
一般注記	1 Introduction -- §1 Historical Remarks -- §2 Background Material -- §3 h-Principles -- §4 The Approximation Problem -- 2 Convex Hulls -- §1 Contractible Spaces of Surrounding Loops -- §2 C-Structures for Relations in Affine Bundles -- §3 The Integral Representation Theorem -- 3 Analytic Theory -- §1 The One-Dimensional Theorem -- §2 The C?-Approximation Theorem -- 4 Open Ample Relations in Spaces of 1-Jets -- §1 C°-Dense h-Principle -- §2 Examples -- 5 Microfibrations -- §1 Introduction -- §2 C-Structures for Relations over Affine Bundles -- §3 The C?-Approximation Theorem -- 6 The Geometry of Jet spaces -- §1 The Manifold X? -- §2 Principal Decompositions in Jet Spaces -- 7 Convex Hull Extensions -- §1 The Microfibration Property -- §2 The h-Stability Theorem -- 8 Ample Relations -- §1 Short Sections -- §2 h-Principle for Ample Relations -- §3 Examples -- §4 Relative h-Principles -- 9 Systems of Partial Differential Equations -- §1 Underdetermined Systems -- §2 Triangular Systems -- §3 C1-Isometric Immersions -- 10 Relaxation Theorem -- §1 Filippov’s Relaxation Theorem -- §2 C?-Relaxation Theorem -- References -- Index of Notation §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 Accessibility summary: This PDF is not accessible. It is based on scanned pages and does not support features such as screen reader compatibility or described non-text content (images, graphs etc). However, it likely supports searchable and selectable text based on OCR (Optical Character Recognition). Users with accessibility needs may not be able to use this content effectively. Please contact us at accessibilitysupport@springernature.com if you require assistance or an alternative format Inaccessible, or known limited accessibility No reading system accessibility options actively disabled Publisher contact for further accessibility information: accessibilitysupport@springernature.com HTTP:URL=https://doi.org/10.1007/978-3-0348-8940-7

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