Convex Integration Theory
Solutions to the h-principle in geometry and topology
Seiten
2010
|
1. Reprint of the 1998 Edition
Springer Basel (Verlag)
978-3-0348-0059-4 (ISBN)
Springer Basel (Verlag)
978-3-0348-0059-4 (ISBN)
This is a comprehensive study of convex integration theory in immersion-theoretic topology, providing methods for solving the h-principle for a variety of problems in differential geometry and topology, with application to PDE theory and optimal control theory.
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.
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.
David Spring is a Professor of mathematics at the Glendon College in Toronto, Canada.
Convex Hulls.- Analytic Theory.- Open Ample Relations in 1-Jet Spaces.- Microfibrations.- The Geometry of Jet Spaces.- Convex Hull Extensions.- Ample Relations.- Systems of Partial Differential Equations.- Relaxation Theory.
Erscheint lt. Verlag | 9.12.2010 |
---|---|
Reihe/Serie | Modern Birkhäuser Classics |
Zusatzinfo | VIII, 213 p. |
Verlagsort | Basel |
Sprache | englisch |
Maße | 155 x 235 mm |
Gewicht | 344 g |
Themenwelt | Mathematik / Informatik ► Mathematik ► Allgemeines / Lexika |
Mathematik / Informatik ► Mathematik ► Analysis | |
Schlagworte | convex integration • Differential Geometry • Equation • Function • Geometry • hull extensions • Integralrechnung • manifold • partial differential equation • Proof • Theorem • Topology |
ISBN-10 | 3-0348-0059-2 / 3034800592 |
ISBN-13 | 978-3-0348-0059-4 / 9783034800594 |
Zustand | Neuware |
Haben Sie eine Frage zum Produkt? |
Mehr entdecken
aus dem Bereich
aus dem Bereich
ein Übungsbuch für Fachhochschulen
Buch | Hardcover (2023)
Carl Hanser (Verlag)
16,99 €