Geometric Structures on Spaces of Weighted Submanifolds
In this paper we use a diffeo-geometric framework based on manifolds that are locally modeled on ''convenient'' vector spaces to study the geometry of some infinite dimensional spaces. Given a finite dimensional symplectic manifold (M,ω), we construct a weak symplectic structure...
Збережено в:
| Видавець: | Інститут математики НАН України |
|---|---|
| Дата: | 2009 |
| Автор: | |
| Формат: | Стаття |
| Мова: | English |
| Опубліковано: |
Інститут математики НАН України
2009
|
| Назва видання: | Symmetry, Integrability and Geometry: Methods and Applications |
| Онлайн доступ: | http://dspace.nbuv.gov.ua/handle/123456789/149103 |
| Теги: |
Додати тег
Немає тегів, Будьте першим, хто поставить тег для цього запису!
|
| Цитувати: | Geometric Structures on Spaces of Weighted Submanifolds / B. Lee // Symmetry, Integrability and Geometry: Methods and Applications. — 2009. — Т. 5. — Бібліогр.: 23 назв. — англ. |
Репозиторії
Digital Library of Periodicals of National Academy of Sciences of Ukraine| id |
irk-123456789-149103 |
|---|---|
| record_format |
dspace |
| spelling |
irk-123456789-1491032019-02-20T01:25:36Z Geometric Structures on Spaces of Weighted Submanifolds Lee, B. In this paper we use a diffeo-geometric framework based on manifolds that are locally modeled on ''convenient'' vector spaces to study the geometry of some infinite dimensional spaces. Given a finite dimensional symplectic manifold (M,ω), we construct a weak symplectic structure on each leaf Iw of a foliation of the space of compact oriented isotropic submanifolds in M equipped with top degree forms of total measure 1. These forms are called weightings and such manifolds are said to be weighted. We show that this symplectic structure on the particular leaves consisting of weighted Lagrangian submanifolds is equivalent to a heuristic weak symplectic structure of Weinstein [Adv. Math. 82 (1990), 133-159]. When the weightings are positive, these symplectic spaces are symplectomorphic to reductions of a weak symplectic structure of Donaldson [Asian J. Math. 3 (1999), 1-15] on the space of embeddings of a fixed compact oriented manifold into M. When M is compact, by generalizing a moment map of Weinstein we construct a symplectomorphism of each leaf Iw consisting of positive weighted isotropic submanifolds onto a coadjoint orbit of the group of Hamiltonian symplectomorphisms of M equipped with the Kirillov-Kostant-Souriau symplectic structure. After defining notions of Poisson algebras and Poisson manifolds, we prove that each space Iw can also be identified with a symplectic leaf of a Poisson structure. Finally, we discuss a kinematic description of spaces of weighted submanifolds. 2009 Article Geometric Structures on Spaces of Weighted Submanifolds / B. Lee // Symmetry, Integrability and Geometry: Methods and Applications. — 2009. — Т. 5. — Бібліогр.: 23 назв. — англ. 1815-0659 2000 Mathematics Subject Classification: 58B99 http://dspace.nbuv.gov.ua/handle/123456789/149103 en Symmetry, Integrability and Geometry: Methods and Applications Інститут математики НАН України |
| institution |
Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| collection |
DSpace DC |
| language |
English |
| description |
In this paper we use a diffeo-geometric framework based on manifolds that are locally modeled on ''convenient'' vector spaces to study the geometry of some infinite dimensional spaces. Given a finite dimensional symplectic manifold (M,ω), we construct a weak symplectic structure on each leaf Iw of a foliation of the space of compact oriented isotropic submanifolds in M equipped with top degree forms of total measure 1. These forms are called weightings and such manifolds are said to be weighted. We show that this symplectic structure on the particular leaves consisting of weighted Lagrangian submanifolds is equivalent to a heuristic weak symplectic structure of Weinstein [Adv. Math. 82 (1990), 133-159]. When the weightings are positive, these symplectic spaces are symplectomorphic to reductions of a weak symplectic structure of Donaldson [Asian J. Math. 3 (1999), 1-15] on the space of embeddings of a fixed compact oriented manifold into M. When M is compact, by generalizing a moment map of Weinstein we construct a symplectomorphism of each leaf Iw consisting of positive weighted isotropic submanifolds onto a coadjoint orbit of the group of Hamiltonian symplectomorphisms of M equipped with the Kirillov-Kostant-Souriau symplectic structure. After defining notions of Poisson algebras and Poisson manifolds, we prove that each space Iw can also be identified with a symplectic leaf of a Poisson structure. Finally, we discuss a kinematic description of spaces of weighted submanifolds. |
| format |
Article |
| author |
Lee, B. |
| spellingShingle |
Lee, B. Geometric Structures on Spaces of Weighted Submanifolds Symmetry, Integrability and Geometry: Methods and Applications |
| author_facet |
Lee, B. |
| author_sort |
Lee, B. |
| title |
Geometric Structures on Spaces of Weighted Submanifolds |
| title_short |
Geometric Structures on Spaces of Weighted Submanifolds |
| title_full |
Geometric Structures on Spaces of Weighted Submanifolds |
| title_fullStr |
Geometric Structures on Spaces of Weighted Submanifolds |
| title_full_unstemmed |
Geometric Structures on Spaces of Weighted Submanifolds |
| title_sort |
geometric structures on spaces of weighted submanifolds |
| publisher |
Інститут математики НАН України |
| publishDate |
2009 |
| url |
http://dspace.nbuv.gov.ua/handle/123456789/149103 |
| citation_txt |
Geometric Structures on Spaces of Weighted Submanifolds / B. Lee // Symmetry, Integrability and Geometry: Methods and Applications. — 2009. — Т. 5. — Бібліогр.: 23 назв. — англ. |
| series |
Symmetry, Integrability and Geometry: Methods and Applications |
| work_keys_str_mv |
AT leeb geometricstructuresonspacesofweightedsubmanifolds |
| first_indexed |
2023-05-20T17:32:09Z |
| last_indexed |
2023-05-20T17:32:09Z |
| _version_ |
1796153520426057728 |