Symplectic Geometry of Integrable Hamiltonian Systems
Springer Basel (Verlag)
978-3-7643-2167-3 (ISBN)
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).
Michèle Audin; Professor of Mathematics at IRMA, Université de Strasbourg et CNRS, France.
A Lagrangian Submanifolds.- I Lagrangian and special Lagrangian immersions in C".- II Lagrangian and special Lagrangian submanifolds in symplectic and Calabi-Yau manifolds.- B Symplectic Toric Manifolds.- I Symplectic Viewpoint.- II Algebraic Viewpoint.- C Geodesic Flows and Contact Toric Manifolds.- I From toric integrable geodesic flows to contact toric manifolds.- II Contact group actions and contact moment maps.- III Proof of Theorem I.38.- List of Contributors.
"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
| Erscheint lt. Verlag | 24.4.2003 |
|---|---|
| Reihe/Serie | Advanced Courses in Mathematics - CRM Barcelona |
| Zusatzinfo | X, 226 p. |
| Verlagsort | Basel |
| Sprache | englisch |
| Maße | 170 x 244 mm |
| Gewicht | 441 g |
| Themenwelt | Mathematik / Informatik ► Mathematik ► Geometrie / Topologie |
| Naturwissenschaften ► Physik / Astronomie | |
| Schlagworte | Contact Geometry • Differentialgeometrie • Differential Geometry • Differenzialgeometrie • Integrable Systems • manifold • Symplectic Geometry |
| ISBN-10 | 3-7643-2167-9 / 3764321679 |
| ISBN-13 | 978-3-7643-2167-3 / 9783764321673 |
| Zustand | Neuware |
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