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Multidimensional Poisson brackets of hydrodynamic type and flat pencils of metrics. (English) Zbl 1170.37027
The multidimensional Poisson brackets of hydrodynamic type are studied. A classification of all nonsingular nondegenerate of such brackets is given for any number $$N$$ of components and for any dimension $$n$$ by differential geometric methods. It is proven that this problem is equivalent to the classification of a special class of compatible flat pencils of metrics. In contrast to one-dimensional case, a nondegenerate multidimensional Poisson bracket of hydrodynamic type cannot be reduced to a constant form by a local change of coordinates. however, it is shown that, for any such bracket, there exists special local coordinates that the bracket is linear wit respect to the fields. The corresponding Poisson brackets are generated by special infinite-dimensional Lie algebras and certain two-cocycles defined on these algebras. One important result obtained by author is that the problem of the description of compatible one-dimensional Poisson of hydrodynamic type is equivalent to that of flat pencils of compatible metrics. The nonlinear equations describing all nonsingular pairs of compatible flat metrics have been previously obtained by the author and used to solve the problem of classification of all multidimensional Poisson brackets. The integrability of these equations was proved by the inverse scattering method. An explicit integration and linearization which reduces the nonlinear case to the solution of linear problems was also obtained.

##### MSC:
 37K05 Hamiltonian structures, symmetries, variational principles, conservation laws (MSC2010) 35F20 Nonlinear first-order PDEs