论文标题
Nijenhuis几何的应用II：多汉米尔顿的最大铅笔流体动力类型
Applications of Nijenhuis geometry II: maximal pencils of multihamiltonian structures of hydrodynamic type
论文作者
论文摘要
我们连接两个先验无关的主题，差异几何形状中的地质等效指标的理论，以及数学物理学中流体动力类型的兼容无限尺寸泊松托架的理论。也就是说，我们证明了一对地理上等效的指标，使一个指标会产生一对括号。我们为这些支架和相应的通勤流构建卡西米尔。有两种方法可以从一对地球上等效的指标中产生大型兼容的泊松结构家族。这些家庭之一是$（n+1）（n+2）/2 $尺寸；我们完全描述它，并表明它是最大的。另一个具有尺寸$ \ le n+2 $，在某种意义上是多项式。我们表明，兼容泊松结构的非平凡多项式家族$ n+2 $是唯一的，来自一对地理上等效的指标。此外，我们将Sinjukov（1961）的结果从恒定的曲率指标转变为任意的爱因斯坦指标。
We connect two a priori unrelated topics, theory of geodesically equivalent metrics in differential geometry, and theory of compatible infinite dimensional Poisson brackets of hydrodynamic type in mathematical physics. Namely, we prove that a pair of geodesically equivalent metrics such that one is flat produces a pair of such brackets. We construct Casimirs for these brackets and the corresponding commuting flows. There are two ways to produce a large family of compatible Poisson structures from a pair of geodesically equivalent metrics one of which is flat. One of these families is $(n+1)(n+2)/2$ dimensional; we describe it completely and show that it is maximal. Another has dimension $\le n+2$ and is, in a certain sense, polynomial. We show that a nontrivial polynomial family of compatible Poisson structures of dimension $n+2$ is unique and comes from a pair of geodesically equivalent metrics. In addition, we generalise a result of Sinjukov (1961) from constant curvature metrics to arbitrary Einstein metrics.
