Dubrovin-Novikov type Poisson brackets (DN-brackets) (Q1119937)

The paper is concerned with canonical forms of the Poisson bracket \[ \{u^ i(x),u^ j(y)\}=\sum g^{ij\alpha}(u(x))\partial \delta (x- y)/\partial x^{\alpha}+\sum (\partial u^ k(x\quad)/\partial x^{\alpha})b_ k^{ij\alpha}(u(x))\delta (x-y) \] of hydrodynamical type (where \(g^{ij\alpha}(u)\), \(u(u^ 1,...,u^ N)\), are nondegenerate metric tensors, \(b_ k^{ij\alpha}=\sum g^{is\alpha} \Gamma^{j\alpha}_{sk}\), and the symmetric connections \(\Gamma^{j\alpha}_{ik}\) related to the metrics are of zero curvature) correcting some previous results presented in several other articles.