By O I Mokhov
This overview provides the differential-geometric concept of homogeneous constructions (mainly Poisson and symplectic structures)on loop areas of soft manifolds, their normal generalizations and functions in mathematical physics and box idea.
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Extra resources for Symplectic and Poisson geometry on loop spaces of smooth manifolds, and integrable equations
Sample text
146) is uniquely defined by on the almost symplectic manifold (M, gij). It is important to note an arbitrary symplectic connection that the class of symplectic connections on almost symplectic manifolds is very rich. 153) on its loop space. On symplectic manifolds there exists a special class of symmetric symplectic connections satisfying the condition It is easy to show that symmetric symplectic connections exist on (M, gij) if and only if (M, gij) is a symplectic manifold, that is, (dg)ijk=0.
3) (but, of course, this set of relations is not sufficient for the classification of such Poisson structures or corresponding infinite-dimensional Lie algebras). 1 An arbitrary non-degenerate multidimensional local Poisson structure of hydrodynamic type is defined by an infinitedimensional Lie algebra of special type and a 2-cocycle on this Lie algebra: The corresponding 2-cocycles on the Lie algebra must have the following form: Let us mention briefly here a general scheme that goes back to Sophus Lie (see [69]) concerning interconnections between infinitedimensional Lie algebras and Poisson structures whose coefficients depend linearly (possibly, non-homogeneously) on the variables ui(x) and their derivatives where In what follows we shall often use this scheme in different situations.
Such systems naturally arise not only in Euler hydrodynamics and gas dynamics but also in the theory of N-layer flows (the Benney equations), as dispersionless limits of different physical systems, as the result of averaging by the Whitham method, and so on. As a rule, systems of hydrodynamic type arising in physically interesting cases are Hamiltonian. 1) was proposed by Dubrovin and Novikov [34], [35]. 1). 1) are preserved under local changes ui=ui(v) of coordinates on a manifold M, and the co efficients of the bracket are transformed as differential-geometric objects on M.