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

*(English. Russian original)*Zbl 0937.53001
Russ. Math. Surv. 53, No. 3, 515-622 (1998); translation from Usp. Mat. Nauk 53, No. 3, 85-192 (1998).

It is well known that the infinite-dimensional generalization of the notion of a symplectic structure is much richer than its finite-dimensional counterpart. In the finite-dimensional case, the Darboux theorem describes the local structure of a symplectic manifold completely. The related infinite dimensional assertion is not valid in general. In order to use these notions in the theory of PDEs, especially integrable ones, to obtain results rich in content, the theory became to study definite types of local Poisson structures on a functional space (depending on functions and its derivatives up to finite order). This study was initiated by Novikov and Dubrovin.

This overview presents a contemporary state-of-art of the field, mainly its differential-geometrical aspects. Various applications of the developed methods to physically significant models, such as equations of associativity in the two-dimensional field theory, two-dimensional nonlinear sigma-models with torsion, etc, are indicated. A separate topic is an important theorem about the hamiltonicity of the restriction of an evolution system to the set of stationary points of its integral; the proof of the theorem is given.

This overview presents a contemporary state-of-art of the field, mainly its differential-geometrical aspects. Various applications of the developed methods to physically significant models, such as equations of associativity in the two-dimensional field theory, two-dimensional nonlinear sigma-models with torsion, etc, are indicated. A separate topic is an important theorem about the hamiltonicity of the restriction of an evolution system to the set of stationary points of its integral; the proof of the theorem is given.

Reviewer: Lev Lerman (Berlin)

##### MSC:

53-02 | Research exposition (monographs, survey articles) pertaining to differential geometry |

37K30 | Relations of infinite-dimensional Hamiltonian and Lagrangian dynamical systems with infinite-dimensional Lie algebras and other algebraic structures |

53D05 | Symplectic manifolds (general theory) |