Algebraic Nijenhuis operators and Kronecker Poisson pencils

Kronecker webs, Nijenhuis operators, and nonlinear PDEs

The aim of this paper is two-fold. First, a survey of the theory of Kronecker webs and their relations with bihamiltonian structures and PDEs is presented. Second, a partial solution to the problem of bisymplectic realization of a bihamiltonian structure is given. Both the goals are achieved by means of the notion of a partial Nijenhuis operator, which is studied in detail.

Nijenhuis几何的应用 III：Frobenius 铅笔与相容的非齐次 Poisson 结构

Applications of Nijenhuis geometry III: Frobenius pencils and compatible non-homogeneous Poisson structures

本文研究含非退化 Darboux 三阶项的非齐次局部 Poisson 结构的相容性，将其转化为 Frobenius 铅笔分类，并用有根内向森林和 Nijenhuis 铅笔给出完整刻画。

Applications of Nijenhuis geometry II: maximal pencils of multihamiltonian structures of hydrodynamic type

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+…

Projections of Jordan bi-Poisson structures that are Kronecker, diagonal actions, and the classical Gaudin systems

We propose a method of constructing completely integrable systems based on reduction of bihamiltonian structures. More precisely, we give an easily checkable necessary and sufficient conditions for the micro-kroneckerity of the reduction (performed with respect to a special type action of a Lie group) of micro-Jordan bihamiltonian structures whose Nijenhuis tensor has constant eigenvalues. The method is applied to the diagonal action of a Lie group $G$ on a direct product of $N$ coadjoint orbits $Ø=O_1\times...\times O_N$ endowed with a bihamiltonian structure whose first generator is the stan…

余秩二奇数秩分布及其最大第一Kronecker指标的h-原理

H-principle for corank two distributions of odd rank and maximal first Kronecker index

本文研究余秩二奇数秩分布，证明其满足多参数C0-接近h-原理，并给出流形允许此类分布的完全拓扑刻画，推广了接触与偶接触情形。