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. This book serves as an introduction to symplectic and contact geometry for graduate students, exploring the underlying geometry of integrable Hamiltonian systems. Includes exercises designed to complement the expositiont, and up-to-date references.

Contenu

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.