The Classification of Nonsingular Multidimensional Dubrovin–Novikov Brackets

In this paper, the well-known Dubrovin–Novikov problem posed as long ago as in 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 Poisson brackets are generated by nontrivial canonical special infinite-dimensional Lie algebras. 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 is played by the theory of compatible metrics earlier constructed by the present author.