Semi-simple generalized Nijenhuis operators

Hassan Boualem; Robert Brouzet

doi:10.3934/jgm.2012.4.385

Article Contents

2012, Volume 4, Issue 4: 385-395. Doi: 10.3934/jgm.2012.4.385

This issue Previous Article Sobolev metrics on shape space, II: Weighted Sobolev metrics and almost local metrics Next Article Invariant forms and automorphisms of locally homogeneous multisymplectic manifolds

Semi-simple generalized Nijenhuis operators

Hassan Boualem^1, and
Robert Brouzet^2,

1.
Univ. Montpellier 2, I3M UMR CNRS 5149, F-34095 Montpellier
2.
Univ. Perpignan Via Domitia, LAboratoire de Mathmatiques et PhySique, EA 4217, F-66860 Perpignan

Received: December 2011

Revised: June 2012

Published: December 2012

Abstract / Introduction Related Papers Cited by

Abstract

We study a special class of endomorphism fields of the generalized tangent bundle ${\mathcal{T}}M:=TM\oplus T^*M$ of a smooth manifold $M$. An operator of this class is defined as follows: it has a vanishing Courant-Nijenhuis torsion and is diagonalizable (after a possible extension of scalars) with constant dimensions of its eigenspaces. Such an endomorphism field is called a semi-simple generalized Nijenhuis operator. The generalized paracomplex and complex structures give examples of such operators.
In this study, we distinguish two cases according to whether the operator has exactly two eigenvalues or has at least three elements in its spectrum. In the first case, we prove that either the operator is affinely related to a generalized complex structure, or it is equivalent to a pair of transverse Dirac structures on ${\mathcal{T}}M$. In the second case, the semi-simple generalized Nijenhuis operator is conjugate to a special kind of generalized Nijenhuis operator obtained from usual Nijenhuis tensors.

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Mathematics Subject Classification: 53D05, 53D17, 53D18, 53A45, 53C15, 53C56.

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