Symplectic Geometry of Integrable Hamiltonian Systems

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).

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

Auteur
Michèle Audin; Professor of Mathematics at IRMA, Université de Strasbourg et CNRS, France.

Texte du rabat

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).

Contenu
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.