The Lie algebroid [10] is a generalization of both concepts of Lie algebra and integrable distribution, being a vector bundle (E, π, M) with a Lie bracket on his space of sections with properties very similar to those of a tangent bundle. The Poisson manifolds are the smooth manifolds equipped with a Poisson bracket on their ring of functions. I have to remark that the cotangent bundle of a Poi...

A Kähler-Nijenhuis manifold is a Kähler manifold M , with metric g, complex structure J and Kähler form Ω, endowed with a Nijenhuis tensor field A that is compatible with the Poisson structure defined by Ω in the sense of the theory of Poisson-Nijenhuis structures. If this happens, and if AJ = ±JA, M is foliated by im A into non degenerate Kähler-Nijenhuis submanifolds. If A is a non degenerate...

This work is the first, and main, of a series papers in progress dedicated to Nijenhuis operators, i.e., fields endomorphisms with vanishing torsion. It serves as an introduction Geometry that should be understood much wider context than before: from local description at generic points singularities global analysis. The goal present paper introduce terminology, develop new important techniques ...

We introduce Courant 1-derivations, which describe a compatibility between algebroids and linear (1,1)-tensor fields lead to the notion of Courant-Nijenhuis algebroids. provide examples 1-derivations on exact show that holomorphic can be viewed as special types By considering Dirac structures, one recovers Dirac-Nijenhuis structures previously studied by authors (in case standard algebroid) obt...

The Nijenhuis tensor Φ = θ1θ −1 0 has n distinct eigenvalues μ = (μ1, . . . , μn) [3]. One can construct a canonical transformation (q, p) 7→ (μ, ν) ((μ, ν) referred to as the Nijenhuis coordinates) and the FDIHS in the Nijenhuis coordinates is separable. Several QBH systems are presented and some relationship between BH and QBH structure is discussed in [1, 2, 4, 5]. The aims of this paper is ...

Two quasi–biHamiltonian systems with three and four degrees of freedom are presented. These systems are shown to be separable in terms of Nijenhuis coordinates. Moreover the most general Pfaffian quasi-biHamiltonian system with an arbitrary number of degrees of freedom is constructed (in terms of Nijenhuis coordinates) and its separability is proved.

In this paper we review the recently proposed path-integral counterpart of the Koopman-von Neumann operatorial approach to classical Hamiltonian mechanics. We identify in particular the geometrical variables entering this tbrmulation and show that they are essentially a basis of the cotangent bundle to the tangent bundle to phase-space. In this space we introduce an extended Poisson brackets st...

Contractions of Leibniz algebras and Courant algebroids by means of (1,1)-tensors are introduced and studied. An appropriate version of Nijenhuis tensors leads to natural deformations of Dirac structures and Lie bialgebroids. One recovers presymplectic-Nijenhuis structures, PoissonNijenhuis structures, and triangular Lie bialgebroids as particular examples. MSC 2000: Primary 17B99; Secondary 17...