Nijenhuis Geometry: singular points and applications

Nijenhuis 几何：奇异点和应用

• 批准号: 529233771
• 负责人: Professor Dr. Vladimir Matveev
• 金额: --
• 依托单位: Lehrstuhl für Differentialgeometrie
• 依托单位国家: 德国
• 项目类别: Research Grants
• 资助国家: 德国
• 项目状态: 未结题
• 来源: https://gepris.dfg.de/gepris/projekt/529233771?language=en
• 关键词: Nijenhuis; Geometry; singular; points; applications

Our long-term goal is to create a new branch of mathematics, Nijenhuis Geometry, with many applications in differential geometry and mathematical physics. Nijenhuis geometry studies Nijenhuis operators, i.e., fields of endomorphisms (or (1,1)-tensor fields) on a smooth manifold such the Nijenhuis torsion vanishes. The project consists of two parts, general theory of Nijenhuis operators and their applications. The main objectives of the first part are: (A) Local description: to what ‘normal’ form can one bring a Nijenhuis operator near every or almost every point by a local coordinate change? (B) Singular points: what does it mean for a point to be generic or singular in the context of Nijenhuis geometry? What singularities are typical? Non-degenerate? Stable? How do Nijenhuis operators behave near non-degenerate and stable singular points? (C) Global properties: what restrictions on a Nijenhuis operator are imposed by compactness of the manifold? Conversely, what are topological obstructions for a manifold carrying a Nijenhuis operator with specific properties (e.g. with no singular points, or with singular points of a prescribed type)? For the second “application’’ part, we selected the following three topics where Nijenhuis operators naturally appear and where our previously obtained results and the results obtained within this project are expected to help: a. Geodesically equivalent metrics. b. Infinite dimensional integrable systems of hydrodynamic type. c. Compatible infinite-dimensional Poisson brackets and related multicomponent integrable systems.

我们的长期目标是创建一个新的数学分支--日本几何，在微分几何和数学物理中有许多应用。Nijenhuis几何研究Nijenhuis算子，即光滑流形上的自同态场(或(1，1)-张量场)，使得Nijenhuis挠率为零。本课题由两部分组成，即Nijenhuis算子的一般理论及其应用。第一部分的主要目标是：(A)局部描述：通过局部坐标变化，可以使Nijenhuis算子接近或几乎接近每个点，其形式是什么？(B)奇点：在Nijenhuis几何学的背景下，一个点是通用的还是奇异的意味着什么？什么是典型的奇点？不堕落？稳定吗？Nijenhuis算子在非退化稳定奇点附近的行为如何？(C)整体性质：流形的紧性对Nijenhuis算子施加了什么限制？反之，对于具有特定性质(例如，没有奇点或具有指定类型的奇点)的Nijenhuis算子的流形来说，什么是拓扑障碍？对于第二个“应用”部分，我们选择了以下三个主题，在这些主题中，Nijenhuis算子自然出现，我们以前获得的结果和本项目中获得的结果有望提供帮助：a.大地测量等价度量。B.无限维水动力可积系统。C.相容无限维Poisson括号及相关的多分量可积系统。