Spectra and symmetric eigentensors of the Lichnerowicz Laplacian on P^n(\comp)

We compute the eigenvalues with multiplicities of the Lichnerowicz Laplacian acting on the space of complex symmetric covariant tensor fields on the complex projective space P^n(\comp). The spaces of symmetric eigentensors are explicitly given

Solutions of the classical Yang-Baxter equation and noncommutative deformations

In this paper, I will show that, if a Lie algebra \G acts on a manifold $P$, any solution of the classical Yang-Baxter equation on \G gives arise to a Poisson tensor on $P$ and a torsion-free and flat contravariant connection (with respect to the Poisson tensor). Moreover, if the action is locally free, the matacurvature of the above contravariant connection vanishes. This will permit to get a large class of manifolds which satisfy the necessary conditions, presented by Hawkins in math.QA/0504232, to the existence of a noncommutative deformation.Comment: 17 page

Riemannian Geometry of Lie Algebroids

We introduce Riemannian Lie algebroids as a generalization of Riemannian manifolds and we show that most of the classical tools and results known in Riemannian geometry can be stated in this setting. We give also some new results on the integrability of Riemannian Lie algebroids.Comment: typos corrected references adde

On the Riemann-Lie algebras and Riemann-Poisson Lie groups

A Riemann-Lie algebra is a Lie algebra $\cal G$ such that its dual ${\cal G}^*$ carries a Riemannian metric compatible (in the sense introduced by th author in C. R. Acad. Paris, t. 333, S\'erie I, (2001) 763-768) with the canonical linear Poisson sructure of ${\cal G}^*$. The notion of Riemann-Lie algebra has its origins in the study, by the author, of Riemann-Poisson manifolds (see Preprint math.DG/0206102 to appear in Differential Geometry and its Applications). In this paper, we show that, for a Lie group $G$, its Lie algebra $\cal G$ carries a structure of Riemann-Lie algebra iff $G$ carries a flat left-invariant Riemannian metric. We use this characterization to construct a huge number of Riemann-Poisson Lie groups (a Riemann-Poisson Lie group is a Poisson Lie group endowed with a left-invariant Riemannian metric compatible with the Poisson structure).Comment: 17 page

Poisson manifolds with compatible pseudo-metric and pseudo-Riemannian Lie algebras

The notion of Poisson manifold with compatible pseudo-metric was introduced by the author in [1]. In this paper, we introduce a new class of Lie algebras which we call a pseudo-Rieamannian Lie algebras. The two notions are strongly related: we prove that a linear Poisson structure on the dual of a Lie algebra has a compatible pseudo-metric if and only if the Lie algebra is a pseudo-Riemannian Lie algebra, and that the Lie algebra obtained by linearizing at a point a Poisson manifold with compatible pseudo-metric is a pseudo-Riemannian Lie algebra. Furthermore, we give some properties of the symplectic leaves of such manifolds, and we prove that every Poisson manifold with compatible metric (every Riemann-Lie algebra) is unimodular. As a final, we classify all pseudo-Riemannian Lie algebras of dimension 2 and 3.Comment: 13 page

Poisson structures compatible with the canonical metric on \reel^3

In this Note, we will characterize the Poisson structures compatible with the canonical metric of \reel^3. We will also give some relvant examples of such structures. The notion of compatibility used in this Note was introduced and studied by the author in previous papers.Comment: 7 page

Killing-Poisson tensors on Riemannian manifolds

We introduce a new class of Poisson structures on a Riemannian manifold. A Poisson structure in this class will be called a Killing-Poisson structure. The class of Killing-Poisson structures contains the class of symplectic structures, the class of Poisson structures studied in{\it (Differential Geometry and its Applications, {\bf Vol. 20, Issue 3} (2004), 279--291)} and the class of Poisson structures induced by some infinitesimal Lie algebras actions on Riemannian manifolds. We show that some classical results on symplectic manifolds (the integrability of the Lie algebroid structure associated to a symplectic structure, the non exactness of a symplectic structure on a compact manifold) remain valid for regular Killing-Poisson structures.Comment: 21 pages, modified version, submitte

The geometry of generalized Cheeger-Gromoll metrics on the total space of transitive Euclidean Lie algebroids

Natural metrics (Sasaki metric, Cheeger-Gromoll metric, Kaluza-Klein metrics etc.. ) on the tangent bundle of a Riemannian manifold is a central topic in Riemannian geometry. Generalized Cheeger-Gromoll metrics is a family of natural metrics $h_{p,q}$ depending on two parameters with $p\in\mathbb{R}$ and $q\geq0$. This family has been introduced recently and possesses interesting geometric properties. If $p=q=0$ we recover the Sasaki metric and when $p=q=1$ we recover the classical Cheeger-Gromoll metric. A transitive Euclidean Lie algebroid is a transitive Lie algebroid with an Euclidean product on its total space. In this paper, we show that natural metrics can be built in a natural way on the total space of transitive Euclidean Lie algebroids. Then we study the properties of generalized Cheeger-Gromoll metrics on this new context. We show a rigidity result of this metrics which generalizes so far all rigidity results known in the case of the tangent bundle. We show also that considering natural metrics on the total space of transitive Euclidean Lie algebroids opens new interesting horizons. For instance, Atiyah Lie algebroids constitute an important class of transitive Lie algebroids and we will show that natural metrics on the total space of Atiyah Euclidean Lie algebroids have interesting properties. In particular, if $M$ is a Riemannian manifold of dimension $n$, then the Atiyah Lie algebroid associated to the $\mathrm{O}(n)$-principal bundle of orthonormal frames over $M$ possesses a family depending on a parameter $k>0$ of transitive Euclidean Lie algebroids structures say $AO(M,k)$. When $M$ is a space form of constant curvature $c$, we show that there exists two constants $C_n<0$ and $K(n,c)>0$ such that $(AO(M,k),h_{1,1})$ is a Riemannian manifold with positive scalar curvature if and only if $c>C_n$ and $0<k\leq K(n,c)$.Comment: Submitte

Solutions of the Yang-Baxter equations on orthogonal groups : the case of oscillator groups

A Lie group is called orthogonal if it carries a bi-invariant pseudo Riemannian metric. Oscillator Lie groups constitutes a subclass of the class of orthogonal Lie groups. In this paper, we determine the Lie bialgebra structures and the solutions of the classical Yang-Baxter equation on a generic class of oscillator Lie groups. On the other hand, we show that any solution of the classical Yang-Baxter equation on an orthogonal Lie group induces a metric in the dual Lie groups associated to this solution. This metric is geodesically complete if and only if the dual are unimodular. More generally, we show that any solution of the generalized Yang-Baxter equation on an orthogonal Lie group determines a left invariant locally symmetric pseudo-Riemannian metric on the corresponding dual Lie groups. Applying this result to oscillator Lie groups we get a large class of solvable Lie groups with flat left invariant Lorentzian metric.Comment: 24 pages,one reference adde

The modular class of a regular Poisson manifold and the Reeb invariant of its symplectic foliation

We show that, for any regular Poisson manifold, there is an injective natural linear map from the first leafwise cohomology space into the first Poisson cohomology space which maps the Reeb class of the symplectic foliation to the modular class of the Poisson manifold. The Riemannian interpretation of those classes will permit us to show that a regular Poisson manifold whose symplectic foliation is of codimension one is unimodular if and only if its symplectic foliation is Riemannian foliation. It permit us also to construct examples of unimodular Poisson manifolds and other which are not unimodular. Finally, we prove that the first leafwise cohomology is an invariant of Morita equivalence.Comment: 11 page