Symplectic and Poisson geometry on loop spaces of smooth manifolds, and integrable equations

Book Cover......Page 1
Dedication......Page 2
Title......Page 3
Contents......Page 4
1 Introduction......Page 7
2.1.1  Complex of skew-symmetric forms on loop spaces of smooth manifolds......Page 10
2.1.2  Symplectic structures......Page 11
2.1.3  Complete description of all local matrix symplectic structures of zero order on loop spaces of smooth manifolds......Page 14
2.1.4  Poisson structures......Page 17
2.1.5  Lagrangian description of local symplectic structures of zero order......Page 22
2.1.6  Compatible Poisson and symplectic structures......Page 23
2.2.1  Homogeneous symplectic forms of the first order......Page 25
2.2.3  Symplectic representation for an arbitrary two-dimensional non-linear sigma-model......Page 28
2.2.4 Examples of bi-Hamiltonian representations for non-linear sigma-models......Page 29
2.3.1  Symplectic representations for degenerate Lagrangian systems......Page 32
2.3.2  Bi-Lagrangian systems......Page 33
2.3.3  Symplectic representation of the Monge-Ampère equation......Page 36
2.4.1  General homogeneous symplectic forms of arbitrary orders......Page 38
2.4.2  Homogeneous symplectic forms of the second order......Page 39
3 Complexes of homogeneous forms on loop spaces of smooth manifolds and their cohomology groups......Page 42
3.2  Complexes of homogeneous forms on loop spaces of smooth manifolds......Page 43
3.3  Cohomology groups of complexes of homogeneous forms on loop spaces of smooth manifolds......Page 45
4.1.1  Multidimensional local Poisson brackets of hydrodynamic type......Page 53
4.1.2 Tensor obstacles for multidimensional local Poisson brackets of hydrodynamic type......Page 56
4.1.3  Infinite-dimensional Lie algebras associated with multidimensional local Poisson brackets of hydrodynamic type......Page 58
4.2.1  One-dimensional homogeneous Hamiltonian systems of hydrodynamic type......Page 60
4.2.2  Non-local Poisson brackets of hydrodynamic type related to metrics of constant Riemannian curvature......Page 62
4.2.3  Further non-local generalizations of Poisson brackets of hydrodynamic type......Page 68
4.3.1  Non-homogeneous local multidimensional Poisson brackets of hydrodynamic type......Page 69
4.3.2  Killing-Poisson bivectors on flat manifolds and Lie—Poisson bivectors......Page 71
4.3.3  Kac-Moody algebras related to non-homogeneous Poisson brackets of hydrodynamic type......Page 72
4.3.4  Reciprocal transformations and non-homogeneous systems of hydrodynamic type......Page 76
4.4.1  Non-local non-homogeneous Poisson brackets of hydrodynamic type......Page 77
4.4.2  The Heisenberg magnet and non-local Poisson structure of hydrodynamic type......Page 78
4.4.3  Killing—Poisson bivectors on spaces of constant Riemannian curvature......Page 79
4.5.1  General Homogeneous Poisson brackets of arbitrary orders......Page 81
4.5.2 Homogeneous Poisson brackets of the second order......Page 82
4.5.3  Homogeneous Poisson brackets of the third order......Page 83
5.1  Equations of associativity as non-diagonalizable integrable homogeneous systems of hydrodynamic type......Page 86
5.2  Poisson and symplectic structures of the equations of associativity......Page 95
5.3  Theorem on a canonical Hamiltonian representation of the restriction of an arbitrary evolution system to the set of stationary points of its non-degenerate integral and its applications to the equations of associativity and systems of hydrodynamic type......Page 108
Acknowledgments......Page 123
References......Page 124
Index......Page 133