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...
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| Published in: | Symmetry, Integrability and Geometry: Methods and Applications |
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| Date: | 2009 |
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| Format: | Article |
| Language: | English |
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Інститут математики НАН України
2009
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| Online Access: | https://nasplib.isofts.kiev.ua/handle/123456789/149103 |
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| Cite this: | Geometric Structures on Spaces of Weighted Submanifolds / B. Lee // Symmetry, Integrability and Geometry: Methods and Applications. — 2009. — Т. 5. — Бібліогр.: 23 назв. — англ. |
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Lee, B. 2019-02-19T17:22:36Z 2019-02-19T17:22:36Z 2009 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 https://nasplib.isofts.kiev.ua/handle/123456789/149103 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. I thank Eckhard Meinrenken for suggesting this project and Yael Karshon who joint supervised this work as part of the author’s PhD thesis. I also thank Boris Khesin for his many helpful suggestions towards improving both the content and exposition of this paper. en Інститут математики НАН України Symmetry, Integrability and Geometry: Methods and Applications Geometric Structures on Spaces of Weighted Submanifolds Article published earlier |
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Digital Library of Periodicals of National Academy of Sciences of Ukraine |
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| title |
Geometric Structures on Spaces of Weighted Submanifolds |
| spellingShingle |
Geometric Structures on Spaces of Weighted Submanifolds Lee, B. |
| 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 |
| author |
Lee, B. |
| author_facet |
Lee, B. |
| publishDate |
2009 |
| language |
English |
| container_title |
Symmetry, Integrability and Geometry: Methods and Applications |
| publisher |
Інститут математики НАН України |
| format |
Article |
| 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.
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| issn |
1815-0659 |
| url |
https://nasplib.isofts.kiev.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. |
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AT leeb geometricstructuresonspacesofweightedsubmanifolds |
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2025-12-07T18:50:23Z |
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2025-12-07T18:50:23Z |
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1850876545204224000 |