Symplectic and Poisson structures on loop spaces of smooth manifolds, and integrable systems

ContentsIntroduction Chapter I. Differential geometry of symplectic structures on loop spaces of smooth manifolds § 1.1. Symplectic and Poisson structures on loop spaces of smooth manifolds. Basic definitions § 1.2. Homogeneous symplectic structures of the first order on loop spaces of pseudo-Riemannian manifolds and two-dimensional non-linear sigma-models with torsion § 1.3. Symplectic and Poisson structures of degenerate Lagrangian systems § 1.4. Homogeneous symplectic structures of the second order on loop spaces of almost symplectic and symplectic manifolds and symplectic connections § 1.5. Complexes of homogeneous forms on loop spaces of smooth manifolds and their cohomology groupsChapter II. Local and non-local Poisson structures of differential-geometric type § 2.1. Riemannian geometry of multidimensional local Poisson structures of hydrodynamic type § 2.2. Hamiltonian systems of hydrodynamic type and metrics of constant Riemannian curvature § 2.3. Non-homogeneous Hamiltonian systems of hydrodynamic type § 2.4. Killing-Poisson bivectors on spaces of constant Riemannian curvature and bi-Hamiltonian structure of the generalized Heisenberg magnets § 2.5. Homogeneous Poisson structures of differential-geometric typeChapter III. The equations of associativity in two-dimensional topological field theory and non-diagonalizable integrable systems of hydrodynamic type § 3.1. Equations of associativity as non-diagonalizable integrable homogeneous systems of hydrodynamic type § 3.2. Poisson and symplectic structures of the equations of associativity § 3.3. 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 Bibliography