The classification of nonsingular multidimensional Dubrovin–Novikov brackets

In this paper the well-known Dubrovin–Novikov problem posed as long ago as 1984 in connection with the Hamiltonian theory of systems of hydrodynamic type, namely, the classification problem for multidimensional Poisson brackets of hydrodynamic type, is solved. In contrast to the one-dimensional case, in the general case, a nondegenerate multidimensional Poisson bracket of hydrodynamic type cannot be reduced to a constant form by a local change of coordinates. Generally speaking, such a Poisson bracket is generated by a nontrivial special infinite-dimensional Lie algebra. In this paper we obtain a classification of all nonsingular nondegenerate multidimensional Poisson brackets of hydrodynamic type for any number N of components and for any dimension n by differential-geometric methods. A key role in the solution of this problem was played by the theory of compatible metrics that had been earlier constructed by the present author. The research was supported by the Max-Planck-Institut für Mathematik (Bonn, Germany), the Russian Science Support Foundation, the Russian Foundation for Basic Research (grants Nos. 0301-00782 and 05-01-00170), and the Program of Support for Leading Scientific Schools (grant No. NSh-4182.2006.1).