Tensor Invariants of the Poisson Brackets of Hydrodynamic Type

Form-invariant solutions for the Poisson brackets of hydrodynamic type on a manifold Mn with (2,0)-tensor gij(u) of rank m ≤ n are derived. Tensor invariants of the Poisson brackets are introduced that include a vector field V (or dynamical system V) on Mn, the Lie derivative LV gij and symmetric (k, 0)-tensors \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$h^{ij\cdots\ell}$$\end{document}. Several scalar invariants of the Poisson brackets are defined. A nilpotent Lie algebra structure is disclosed in the space of 1-forms \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathcal{A}}_u \subset T^*_u(M^n)$$\end{document} that annihilate the (2,0)-tensor gij(u). Applications to the one-dimensional gas dynamics are presented.