Geometric Theory of Foliations:
Gespeichert in:
| 1. Verfasser: | |
|---|---|
| Format: | Elektronisch E-Book |
| Sprache: | English |
| Veröffentlicht: |
Boston, MA
Birkhäuser Boston
1985
|
| Schlagworte: | |
| Online-Zugang: | Volltext |
| Beschreibung: | Intuitively, a foliation corresponds to a decomposition of a manifold into a union of connected, disjoint submanifolds of the same dimension, called leaves, which pile up locally like pages of a book. The theory of foliations, as it is known, began with the work of C. Ehresmann and G. Reeb, in the 1940's; however, as Reeb has himself observed, already in the last century P. Painleve saw the necessity of creating a geometric theory (of foliations) in order to better understand the problems in the study of solutions of holomorphic differential equations in the complex field. The development of the theory of foliations was however provoked by the following question about the topology of manifolds proposed by H. Hopf in the 3 1930's: "Does there exist on the Euclidean sphere S a completely integrable vector field, that is, a field X such that X· curl X • 0?" By Frobenius' theorem, this question is equivalent to the following: "Does there exist on the 3 sphere S a two-dimensional foliation?" This question was answered affirmatively by Reeb in his thesis, where he 3 presents an example of a foliation of S with the following characteristics: There exists one compact leaf homeomorphic to the two-dimensional torus, while the other leaves are homeomorphic to two-dimensional planes which accu mulate asymptotically on the compact leaf. Further, the foliation is C"" |
| Beschreibung: | 1 Online-Ressource (VIII, 206 p) |
| ISBN: | 9781461252924 9781468471496 |
| DOI: | 10.1007/978-1-4612-5292-4 |
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| 500 | |a Intuitively, a foliation corresponds to a decomposition of a manifold into a union of connected, disjoint submanifolds of the same dimension, called leaves, which pile up locally like pages of a book. The theory of foliations, as it is known, began with the work of C. Ehresmann and G. Reeb, in the 1940's; however, as Reeb has himself observed, already in the last century P. Painleve saw the necessity of creating a geometric theory (of foliations) in order to better understand the problems in the study of solutions of holomorphic differential equations in the complex field. The development of the theory of foliations was however provoked by the following question about the topology of manifolds proposed by H. Hopf in the 3 1930's: "Does there exist on the Euclidean sphere S a completely integrable vector field, that is, a field X such that X· curl X • 0?" By Frobenius' theorem, this question is equivalent to the following: "Does there exist on the 3 sphere S a two-dimensional foliation?" This question was answered affirmatively by Reeb in his thesis, where he 3 presents an example of a foliation of S with the following characteristics: There exists one compact leaf homeomorphic to the two-dimensional torus, while the other leaves are homeomorphic to two-dimensional planes which accu mulate asymptotically on the compact leaf. Further, the foliation is C"" | ||
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Datensatz im Suchindex
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| author | Camacho, César |
| author_facet | Camacho, César |
| author_role | aut |
| author_sort | Camacho, César |
| author_variant | c c cc |
| building | Verbundindex |
| bvnumber | BV042420382 |
| classification_tum | MAT 000 |
| collection | ZDB-2-SMA ZDB-2-BAE |
| ctrlnum | (OCoLC)860355459 (DE-599)BVBBV042420382 |
| dewey-full | 516 |
| dewey-hundreds | 500 - Natural sciences and mathematics |
| dewey-ones | 516 - Geometry |
| dewey-raw | 516 |
| dewey-search | 516 |
| dewey-sort | 3516 |
| dewey-tens | 510 - Mathematics |
| discipline | Mathematik |
| doi_str_mv | 10.1007/978-1-4612-5292-4 |
| format | Electronic eBook |
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| institution | BVB |
| isbn | 9781461252924 9781468471496 |
| language | English |
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| spelling | Camacho, César Verfasser aut Geometric Theory of Foliations by César Camacho, Alcides Lins Neto Boston, MA Birkhäuser Boston 1985 1 Online-Ressource (VIII, 206 p) txt rdacontent c rdamedia cr rdacarrier Intuitively, a foliation corresponds to a decomposition of a manifold into a union of connected, disjoint submanifolds of the same dimension, called leaves, which pile up locally like pages of a book. The theory of foliations, as it is known, began with the work of C. Ehresmann and G. Reeb, in the 1940's; however, as Reeb has himself observed, already in the last century P. Painleve saw the necessity of creating a geometric theory (of foliations) in order to better understand the problems in the study of solutions of holomorphic differential equations in the complex field. The development of the theory of foliations was however provoked by the following question about the topology of manifolds proposed by H. Hopf in the 3 1930's: "Does there exist on the Euclidean sphere S a completely integrable vector field, that is, a field X such that X· curl X • 0?" By Frobenius' theorem, this question is equivalent to the following: "Does there exist on the 3 sphere S a two-dimensional foliation?" This question was answered affirmatively by Reeb in his thesis, where he 3 presents an example of a foliation of S with the following characteristics: There exists one compact leaf homeomorphic to the two-dimensional torus, while the other leaves are homeomorphic to two-dimensional planes which accu mulate asymptotically on the compact leaf. Further, the foliation is C"" Mathematics Geometry Mathematik Blätterung (DE-588)4007006-2 gnd rswk-swf Geometrie (DE-588)4020236-7 gnd rswk-swf Differentialgeometrie (DE-588)4012248-7 gnd rswk-swf Blätterung (DE-588)4007006-2 s Geometrie (DE-588)4020236-7 s 1\p DE-604 Differentialgeometrie (DE-588)4012248-7 s 2\p DE-604 Lins Neto, Alcides Sonstige oth https://doi.org/10.1007/978-1-4612-5292-4 Verlag Volltext 1\p cgwrk 20201028 DE-101 https://d-nb.info/provenance/plan#cgwrk 2\p cgwrk 20201028 DE-101 https://d-nb.info/provenance/plan#cgwrk |
| spellingShingle | Camacho, César Geometric Theory of Foliations Mathematics Geometry Mathematik Blätterung (DE-588)4007006-2 gnd Geometrie (DE-588)4020236-7 gnd Differentialgeometrie (DE-588)4012248-7 gnd |
| subject_GND | (DE-588)4007006-2 (DE-588)4020236-7 (DE-588)4012248-7 |
| title | Geometric Theory of Foliations |
| title_auth | Geometric Theory of Foliations |
| title_exact_search | Geometric Theory of Foliations |
| title_full | Geometric Theory of Foliations by César Camacho, Alcides Lins Neto |
| title_fullStr | Geometric Theory of Foliations by César Camacho, Alcides Lins Neto |
| title_full_unstemmed | Geometric Theory of Foliations by César Camacho, Alcides Lins Neto |
| title_short | Geometric Theory of Foliations |
| title_sort | geometric theory of foliations |
| topic | Mathematics Geometry Mathematik Blätterung (DE-588)4007006-2 gnd Geometrie (DE-588)4020236-7 gnd Differentialgeometrie (DE-588)4012248-7 gnd |
| topic_facet | Mathematics Geometry Mathematik Blätterung Geometrie Differentialgeometrie |
| url | https://doi.org/10.1007/978-1-4612-5292-4 |
| work_keys_str_mv | AT camachocesar geometrictheoryoffoliations AT linsnetoalcides geometrictheoryoffoliations |