Related papers: A class of Poisson-Nijenhuis structures on a tange…

An associated Nijenhuis tensor of endomorphisms in the tangent bundle is introduced. Special attention is paid to such tensors for an almost hypercomplex structure and the metric of Hermitian-Norden type. There are studied relations between…

A tensor -- meaning here a tensor field $\Theta$ of any type $(p,q)$ on a manifold -- may be called integrable if it is parallel relative to some torsion-free connection. We provide analytical and geometric characterizations of…

We study pairs of structures, such as the Poisson-Nijenhuis structures, on the tangent bundle of a manifold or, more generally, on a Lie algebroid or a Courant algebroid. These composite structures are defined by two of the following, a…

In this paper we prove that both complete and vertical lifts of a Poisson vector field from a Poisson manifold $(M, \pi)$ to its tangent bundle $(TM, \pi_{TM})$ are also Poisson. We use this fact to describe the infinitesimal deformations…

We extend the calculus of multiplicative vector fields and differential forms and their intrinsic derivatives from Lie groups to Lie groupoids; this generalization turns out to include also the classical process of complete lifting from…

We use some natural lifts defined on the cotangent bundle T*M of a Riemannian manifold (M,g) in order to construct an almost Hermitian structure (G,J) of diagonal type. The obtained almost complex structure J on T*M is integrable if and…

Extending earlier work(*), we examine the deformation of the canonical symplectic structure in a cotangent bundle $T^\star(\Q)$ by additional terms implying the Poisson non-commutativity of both configuration and momentum variables. In this…

The notion of Poisson quasi-Nijenhuis manifold generalizes that of Poisson-Nijenhuis manifold. The relevance of the latter in the theory of completely integrable systems is well established since the birth of the bi-Hamiltonian approach to…

We study lift metrics and lift connections on the tangent bundle $TM$ of a Riemannian manifold $(M,g)$. We also investigate the statistical and Codazzi couples of $TM$ and their consequences on the geometry of $M$. Finally, we prove a…

A Nijenhuis operator on a manifold $M$ is a $(1,1)$ tensor $\mathcal N$ whose Nijenhuis-torsion vanishes. A Nijenhuis operator $\mathcal N$ on $M$ determines a Lie algebroid structure $(TM)_{\mathcal N}$ on the tangent bundle $TM$. In this…

We study the conditions under which the cotangent bundle $T^*M$ of a Riemaannian manifold $(M,g)$, endowed with a K\"ahlerian structure $(G,J)$ of general natural lift type (see \cite{Druta1}), is Einstein. We first obtain a general natural…

We introduce Lie-Nijenhuis bialgebroids as Lie bialgebroids endowed with an additional derivation-like object. They give a complete infinitesimal description of Poisson-Nijenhuis groupoids, and key examples include Poisson-Nijenhuis…

In earlier work (*) we studied an extension of the canonical symplectic structure in the cotangent bundle of an affine space ${\cal Q}={\bf R}^N$, by additional terms implying the Poisson non-commutativity of both configuration and momentum…

We consider some natural (functorial) lifts of geometric objects associated with statistical manifolds (metric tensor, dual connections, skewness tensor, etc.) to higher tangent bundles. It turns out that the lifted objects form again a…

We study the conditions under which a K\"ahlerian structure $(G,J)$ of general natural lift type on the cotangent bundle $T^*M$ of a Riemannian manifold $(M,g)$ has constant holomorphic sectional curvature. We obtain that a certain…

We study some properties of the tangent bundles with metrics of general natural lifted type. We consider a Riemannian manifold $(M,g)$ and we find the conditions under which the Riemannian manifold $(TM,G)$, where $TM$ is the tangent bundle…

By studying the Fr\"olicher-Nijenhuis decomposition of cohomology operators (that is, derivations $D$ of the exterior algebra $\Omega (M)$ with $\mathbb{Z}-$degree $1$ and $D^2=0$), we describe new examples of Lie algebroid structures on…

We introduce a notion of noncommutative Poisson-Nijenhuis structure on the path algebra of a quiver. In particular, we focus on the case when the Poisson bracket arises from a noncommutative symplectic form. The formalism is then applied to…

We investigate the structure of smooth holomorphic foliations with numerically flat tangent bundles on compact K\"ahler manifolds. Extending earlier results on non-uniruled projective manifolds by the second and fourth authors, we show that…

A new method to construct Hamiltonian functions in involution is presented. We show that on left-symmetric algebras a Nijenhuis-tensor is given in a natural manner by the usual right-multiplication. Furthermore we prove that symplectic…