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2003 | Buch

Introduction Kapitel lesen Erstes Kapitel lesen

Symplectic Geometry of Integrable Hamiltonian Systems

verfasst von: Michèle Audin, Ana Cannas da Silva, Eugene Lerman

Verlag: Birkhäuser Basel

Buchreihe : Advanced Courses in Mathematics - CRM Barcelona

Enthalten in: Professional Book Archive

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Über dieses Buch

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).

Inhaltsverzeichnis

Frontmatter

Lagrangian Submanifolds

Frontmatter
Introduction
Abstract
This text is an introduction to Lagrangian and special Lagrangian submanifolds. Special Lagrangian submanifolds were invented twenty years ago by Harvey and Lawson [18]. They have become very fashionable recently, after the work of McLean [25], leading to the beautiful speculations of Strominger, Yau and Zaslow [32] and the remarkable papers of Hitchin [19, 20] and Donaldson [11]
Michèle Audin, Ana Cannas da Silva, Eugene Lerman
Chapter I. Lagrangian and special Lagrangian immersions in Cn
Abstract
In this chapter, I define Lagrangian and special Lagrangian immersions in Cn. To begin with, I explain that Cn is the standardrealvector space endowed with a non degenerate alternated bilinear form (§I.1) and use this “symplectic structure” to define Lagrangian subspaces and immersions (§§I.2, I.3 and I.4). Later, I use the complex structure as well, to definespecialLagrangian immersions (§I.5)
Michèle Audin, Ana Cannas da Silva, Eugene Lerman
Chapter II. Lagrangian and special Lagrangian submanifolds in Symplectic and Calabi-Yau manifolds
Abstract
In order to deform a Lagrangian submanifold in Cn, we must understand how a tubular neighbourhood looks like. We prove here that a Lagrangian submanifold has a neighbourhood which is diffeomorphic to a neighbourhood of the zero section in its cotangent bundle. To be precise and explicit, we need to define a symplectic structure on the cotangent bundles and more generally to say what a symplectic structure on a manifold is
Michèle Audin, Ana Cannas da Silva, Eugene Lerman

Symplectic Toric Manifolds

Frontmatter
Chapter I. Symplectic Viewpoint
Abstract
In order to define symplectic toric manifolds, we begin by introducing the basic objects in symplectic/hamiltonian geometry/mechanics which lead to their consideration. Our discussion centers around moment maps
Michèle Audin, Ana Cannas da Silva, Eugene Lerman
Chapter II. Algebraic Viewpoint
Abstract
The goal of this lecture is to explain toric manifolds as a special class of projective varieties. The first five sections contain a crash course on notions and basic facts about algebraic varieties, mostly in order to fix notation. The combinatorial flavor of toric varieties is postponed until Lecture 5
Michèle Audin, Ana Cannas da Silva, Eugene Lerman

Geodesic Flows and Contact Toric Manifolds

Frontmatter
Chapter I. From toric integrable geodesic flows to contact toric manifolds
Abstract
We start with an innocuous sounding problem
Michèle Audin, Ana Cannas da Silva, Eugene Lerman
Chapter II. Contact group actions and contact moment maps
Abstract
Moment maps exist in the category of contact group actions. In fact moment maps exist for all contact actions. This is because a contact form defines a bijection between contact vector fields and smooth functions
Michèle Audin, Ana Cannas da Silva, Eugene Lerman
Chapter III. Proof of Theorem I.38
Abstract
The rest of the lecture notes will be devoted to a proof of Theorem I.38. Right from the beginning the proof will bifurcate into two cases: the contact manifold B is 3-dimensional and dim B > 3. If dimB = 3 we will argue directly using slices that the orbit spaceB/Gis homeomorphic to a closed interval [0, 1] and then use this to compute the integral cohomology ofB. This will show that B cannot be homeomorphic to \( S^* \mathbb{T}^2 = \mathbb{T}^3 \)
Michèle Audin, Ana Cannas da Silva, Eugene Lerman
Backmatter
Metadaten
Titel
Symplectic Geometry of Integrable Hamiltonian Systems
verfasst von
Michèle Audin
Ana Cannas da Silva
Eugene Lerman
Copyright-Jahr
2003
Verlag
Birkhäuser Basel
Electronic ISBN
978-3-0348-8071-8
Print ISBN
978-3-7643-2167-3
DOI
https://doi.org/10.1007/978-3-0348-8071-8