Convex Integration Theory

Solutions to the h-principle in geometry and topology

Buch | Hardcover

VIII, 213 Seiten

1997 | 1998
Springer Basel (Verlag)
978-3-7643-5805-1 (ISBN)

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§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.

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.

Erscheint lt. Verlag	18.12.1997
Reihe/Serie	Monographs in Mathematics
Zusatzinfo	VIII, 213 p. 2 illus.
Verlagsort	Basel
Sprache	englisch
Maße	155 x 235 mm
Gewicht	499 g
Themenwelt	Mathematik / Informatik ► Mathematik ► Allgemeines / Lexika
Themenwelt	Mathematik / Informatik ► Mathematik ► Geometrie / Topologie
Schlagworte	Differential Geometry • Differential topology • Equation • Function • Geometry • manifold • Theorem • Topology
ISBN-10	3-7643-5805-X / 376435805X
ISBN-13	978-3-7643-5805-1 / 9783764358051
Zustand	Neuware