非奇异多维Dubrovin-Novikov括号的分类
The classification of nonsingular multidimensional Dubrovin-Novikov brackets
摘要：
本文解决了早在1984年就提出的与流体力学型系统的Hamilton理论有关的著名的Dubrovin-Novikov问题，即流体力学型多维泊松括号的分类问题。与一维情形相反，在一般情形下，非退化的流体动力学型多维泊松括号不能通过坐标的局部变化而化为常数形式。一般来说，这样的泊松括号是由非平凡的特殊无限维李代数生成的。本文用微分几何方法对任意维数n，任意分量数N的流体动力型非奇异非退化多维泊松括号进行了分类.在这个问题的解决方案中发挥了关键作用的兼容度量的理论，已较早地由本作者构建。
英文摘要：
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.
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