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An involutivity theorem for a class of Poisson quasi-Nijenhuis manifolds

Eber Chuño Vizarreta, Gregorio Falqui, Igor Mencattini, Marco Pedroni

Abstract

This note aims to continue our study about the applications of Poisson quasi-Nijenhuis geometry to the theory of classical completely integrable systems. More precisely, we will present new versions of the deformation and involutivity theorems, under the hypothesis that the closed 2-form triggering the deformation and the closed 3-form defining the Poisson quasi-Nijenhuis structure are factorized. These results will be supplemented by several examples of involutive Poisson quasi- Nijenhuis manifolds.

An involutivity theorem for a class of Poisson quasi-Nijenhuis manifolds

Abstract

This note aims to continue our study about the applications of Poisson quasi-Nijenhuis geometry to the theory of classical completely integrable systems. More precisely, we will present new versions of the deformation and involutivity theorems, under the hypothesis that the closed 2-form triggering the deformation and the closed 3-form defining the Poisson quasi-Nijenhuis structure are factorized. These results will be supplemented by several examples of involutive Poisson quasi- Nijenhuis manifolds.
Paper Structure (4 sections, 8 theorems, 71 equations, 1 figure, 1 table)

This paper contains 4 sections, 8 theorems, 71 equations, 1 figure, 1 table.

Table of Contents

  1. Introduction
  2. Preliminaries on PN structures, PqN structures and their deformations
  3. Deformations with factorized 2-forms
  4. An involutivity theorem

Key Result

Theorem 2

Let $({\mathcal{M}},\pi,N,\phi)$ be a PqN manifold and let $H_k=\frac{1}{2k}\operatorname{Tr} N^k$. Suppose that there exists a 2-form $\Omega$ such that: Then $\{H_j,H_k\}=0$ for all $j,k$.

Figures (1)

  • Figure 1: Deformations from the Das-Okubo PN structure.

Theorems & Definitions (24)

  • Remark 1
  • Theorem 2: Involutivity theorem
  • Theorem 3: Deformation theorem
  • Lemma 4
  • proof
  • Proposition 5
  • proof
  • Remark 6
  • Example 7
  • Remark 8
  • ...and 14 more