Overview
- Expanded lecture notes originating from a summer school at the CRM Barcelona
- Serves as an introduction to symplectic and contact geometry for graduate students
- Explores the underlying (symplectic) geometry of integrable Hamiltonian systems
Part of the book series: Advanced Courses in Mathematics - CRM Barcelona (ACMBIRK)
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Table of contents (8 chapters)
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Front Matter
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Lagrangian Submanifolds
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Front Matter
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Symplectic Toric Manifolds
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Front Matter
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Geodesic Flows and Contact Toric Manifolds
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Front Matter
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Back Matter
Keywords
About this book
Among all the Hamiltonian systems, the integrable ones have special geometric properties; in particular, their solutions are very regular and quasi-periodic. The quasi-periodicity of the solutions of an integrable system is a result of the fact that the system is invariant under a (semi-global) torus action. It is thus natural to investigate the symplectic manifolds that can be endowed with a (global) torus action. This leads to symplectic toric manifolds (Part B of this book). Physics makes a surprising come-back in Part A: to describe Mirror Symmetry, one looks for a special kind of Lagrangian submanifolds and integrable systems, the special Lagrangians. Furthermore, integrable Hamiltonian systems on punctured cotangent bundles are a starting point for the study of contact toric manifolds (Part C of this book).
Reviews
"This book, an expanded version of the lectures delivered by the authors at the 'Centre de Recerca Matemàtica' Barcelona in July 2001, is designed for a modern introduction to symplectic and contact geometry to graduate students. It can also be useful to research mathematicians interested in integrable systems. The text includes up-to-date references, and has three parts. The first part, by Michèle Audin, contains an introduction to Lagrangian and special Lagrangian submanifolds in symplectic and Calabi-Yau manifolds…. The second part, by Ana Cannas da Silva, provides an elementary introduction to toric manifolds (i.e. smooth toric varieties)…. In these first two parts, there are exercises designed to complement the exposition or extend the reader's understanding…. The last part, by Eugene Lerman, is devoted to the topological study of these manifolds."
—ZENTRALBLATT MATH
Authors and Affiliations
Bibliographic Information
Book Title: Symplectic Geometry of Integrable Hamiltonian Systems
Authors: Michèle Audin, Ana Cannas Silva, Eugene Lerman
Series Title: Advanced Courses in Mathematics - CRM Barcelona
DOI: https://doi.org/10.1007/978-3-0348-8071-8
Publisher: Birkhäuser Basel
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eBook Packages: Springer Book Archive
Copyright Information: Springer Basel AG 2003
Softcover ISBN: 978-3-7643-2167-3Published: 24 April 2003
eBook ISBN: 978-3-0348-8071-8Published: 06 December 2012
Series ISSN: 2297-0304
Series E-ISSN: 2297-0312
Edition Number: 1
Number of Pages: X, 226
Topics: Differential Geometry, Manifolds and Cell Complexes (incl. Diff.Topology), Mathematical Methods in Physics