Convex Integration Theory: Solutions to the h-principle in geometry and topology
Gespeichert in:
| 1. Verfasser: | |
|---|---|
| Format: | Elektronisch E-Book |
| Sprache: | English |
| Veröffentlicht: |
Basel
Birkhäuser Basel
1998
|
| Schriftenreihe: | Monographs in Mathematics
92 |
| Schlagworte: | |
| Online-Zugang: | Volltext |
| Beschreibung: | §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 |
| Beschreibung: | 1 Online-Ressource (VIII, 213 p) |
| ISBN: | 9783034889407 9783034898362 |
| DOI: | 10.1007/978-3-0348-8940-7 |
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| discipline | Mathematik |
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| indexdate | 2024-07-10T01:21:10Z |
| institution | BVB |
| isbn | 9783034889407 9783034898362 |
| language | English |
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| spelling | Spring, David Verfasser aut Convex Integration Theory Solutions to the h-principle in geometry and topology by David Spring Basel Birkhäuser Basel 1998 1 Online-Ressource (VIII, 213 p) txt rdacontent c rdamedia cr rdacarrier Monographs in Mathematics 92 §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 Mathematics Mathematics, general Mathematik Differentialtopologie (DE-588)4012255-4 gnd rswk-swf Differentialtopologie (DE-588)4012255-4 s 1\p DE-604 https://doi.org/10.1007/978-3-0348-8940-7 Verlag Volltext 1\p cgwrk 20201028 DE-101 https://d-nb.info/provenance/plan#cgwrk |
| spellingShingle | Spring, David Convex Integration Theory Solutions to the h-principle in geometry and topology Mathematics Mathematics, general Mathematik Differentialtopologie (DE-588)4012255-4 gnd |
| subject_GND | (DE-588)4012255-4 |
| title | Convex Integration Theory Solutions to the h-principle in geometry and topology |
| title_auth | Convex Integration Theory Solutions to the h-principle in geometry and topology |
| title_exact_search | Convex Integration Theory Solutions to the h-principle in geometry and topology |
| title_full | Convex Integration Theory Solutions to the h-principle in geometry and topology by David Spring |
| title_fullStr | Convex Integration Theory Solutions to the h-principle in geometry and topology by David Spring |
| title_full_unstemmed | Convex Integration Theory Solutions to the h-principle in geometry and topology by David Spring |
| title_short | Convex Integration Theory |
| title_sort | convex integration theory solutions to the h principle in geometry and topology |
| title_sub | Solutions to the h-principle in geometry and topology |
| topic | Mathematics Mathematics, general Mathematik Differentialtopologie (DE-588)4012255-4 gnd |
| topic_facet | Mathematics Mathematics, general Mathematik Differentialtopologie |
| url | https://doi.org/10.1007/978-3-0348-8940-7 |
| work_keys_str_mv | AT springdavid convexintegrationtheorysolutionstothehprincipleingeometryandtopology |