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Birkhäuser - Birkhäuser Mathematics | Symplectic Geometry of Integrable Hamiltonian Systems

Symplectic Geometry of Integrable Hamiltonian Systems

Audin, Michèle, Cannas da Silva, Ana, Lerman, Eugene

2003, 236 p.

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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).

Content Level » Graduate

Keywords » Differential Geometry - Integrable Systems - contact geometry - manifold - symplectic geometry

Related subjects » Birkhäuser Mathematics - Birkhäuser Physics

Table of contents 

A Lagrangian Submanifolds.- I Lagrangian and special Lagrangian immersions in C“.- I.1 Symplectic form on C“, symplectic vector spaces.- Ll.a Symplectic vector spaces.- I.l.b Symplectic bases.- I.l.c The symplectic form as a differential form.- I.l.d The symplectic group.- I.l.e Orthogonality, isotropy.- 1.2 Lagrangian subspaces.- I.2.a Definition of Lagrangian subspaces.- I.2.b The symplectic reduction.- 1.3 The Lagrangian Grassmannian.- I.3.a The Grassmannian A“t as a homogeneous space.- I.3.b The manifold An.- I.3.c The tautological vector bundle.- I.3.d The tangent bundle to A“.- I.3.e The case of oriented Lagrangian subspaces.- I.3.f The determinant and the Maslov class.- I.4 Lagrangian submanifolds in Cn.- I.4.a Lagrangian submanifolds described by functions.- I.4.b Wave fronts.- I.4.c Other examples.- I.4.d The Gauss map.- I.5 Special Lagrangian submanifolds in Cn.- I.5.a Special Lagrangian subspaces.- I.5.b Special Lagrangian submanifolds.- I.5.c Graphs of forms.- I.5.d Normal bundles of surfaces.- I.5.e From integrable systems.- I.5.f Special Lagrangian submanifolds invariant under SO(n)..- I.6 Appendices.- I.6.a The topology of the symplectic group.- I.6.b Complex structures.- I.6.c Hamiltonian vector fields, integrable systems.- Exercises.- II Lagrangian and special Lagrangian submanifolds in symplectic and Calabi-Yau manifolds.- II.1 Symplectic manifolds.- II.2 Lagrangian submanifolds and immersions.- II.2.a In cotangent bundles.- I1.3 Tubular neighborhoods of Lagrangian submanifolds.- II.3.a Moser’s method.- II.3.b Tubular neighborhoods.- II.3.c“Moduli space” of Lagrangian submanifolds.- II.4 Calabi-Yau manifolds.- II.4.a Definition of the Calabi-Yau manifolds.- II.4.b Yau’s theorem.- II.4.c Examples of Calabi-Yau manifolds.- II.4.d Special Lagrangian submanifolds.- II.5 Special Lagrangians in real Calabi-Yau manifolds.- II.5.a Real manifolds.- II.5.b Real Calabi-Yau manifolds.- II.5.c The example of elliptic curves 68.- II.5.d Special Lagrangians in real Calabi-Yau manifolds.- 11.6 Moduli space of special Lagrangian submanifolds.- I1.7 Towards mirror symmetry?.- II.7.a Fibrations in special Lagrangian submanifolds 74.- II.7.b Mirror symmetry.- Exercises.- B Symplectic Toric Manifolds.- I Symplectic Viewpoint.- I.1 Symplectic Toric Manifolds.- I.1.1 Symplectic Manifolds.- I.1.2 Hamiltonian Vector Fields.- I.1.3 Integrable Systems.- I.1.4 Hamiltonian Actions.- I.1.5 Hamiltonian Torus Actions.- 1.1.6 Symplectic Toric Manifolds.- I.2 Classification.- 1.2.1 Delzant’s Theorem.- I.2.2 Orbit Spaces.- I.2.3 Symplectic Reduction.- I.2.4 Extensions of Symplectic Reduction.- I.2.5 Delzant’s Construction.- I.2.6 Idea Behind Delzant’s Construction.- I.3 Moment Polytopes.- I.3.1 Equivariant Darboux Theorem.- I.3.2 Morse Theory.- I.3.3 Homology of Symplectic Toric Manifolds.- I.3.4 Symplectic Blow-Up.- I.3.5 Blow-Up of Toric Manifolds.- I.3.6 Symplectic Cutting.- II Algebraic Viewpoint.- II.1 Toric Varieties.- II.1.1 Affine Varieties.- II.1.2 Rational Maps on Affine Varieties.- II.1.3 Projective Varieties.- II.1.4 Rational Maps on Projective Varieties.- II.1.5 Quasiprojective Varieties.- II.1.6 Toric Varieties.- II.2 Classification.- 1I.2.1 Spectra.- II.2.2 Toric Varieties Associated to Semigroups.- I1.2.3 Classification of Affine Toric Varieties.- II.2.4 Fans.- 1I.2.5 Toric Varieties Associated to Fans.- 1I.2.6 Classification of Normal Toric Varieties.- I1.3 Moment Polytopes.- II.3.1 Equivariantly Projective Toric Varieties.- II.3.2 Weight Polytopes.- II.3.3 Orbit Decomposition.- II.3.4 Fans from Polytopes.- II.3.5 Classes of Toric Varieties.- II.3.6 Symplectic vs. Algebraic.- C Geodesic Flows and Contact Toric Manifolds.- I From toric integrable geodesic flows to contact toric manifolds.- I.1 Introduction.- 1.2 Symplectic cones and contact manifolds.- II Contact group actions and contact moment maps.- III Proof of Theorem I.38.- III.1 Homogeneous vector bundles and slices.- III.2 The 3-dimensional case.- III.3 Uniqueness of symplectic toric manifolds.- III3.1 Cecil. cohomology.- III.4 Proof of Theorem I.38, part three.- III.4.1 Morse theory on orbifolds.- List of Contributors.

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