Mohamed Boucetta

Faculté des Sciences et Techniques Marrakech · Department of mathematics

Kundt spacetimes are of great importance to general relativity. We show that a Kundt spacetime is a Lorentz manifold with a non-singular isotropic geodesic vector field having its orthogonal distribution integrable and determining a totally geodesic foliation. We give the local structure of Kundt spacetimes and some properties of left invariant Kun...

Let $G$ be an arbitrary, connected, simply connected and unimodular Lie group of dimension $3$. On the space $\mathfrak{M}(G)$ of left-invariant Lorentzian metrics on $\G$, there exists a natural action of the group $Aut(G)$ of automorphisms of $G$, so it yields an equivalence relation $\backsimeq$ on $\mathfrak{M}(G)$, in the following way: \;$h_1...

Let $(G,\Omega)$ be a symplectic Lie group, i.e, a Lie group endowed with a left invariant symplectic form. If $\G$ is the Lie algebra of $G$ then we call $(\G,\omega=\Om(e))$ a symplectic Lie algebra. The product $\bullet$ on $\G$ defined by $3\omega\left(x\bullet y,z\right)=\omega\left([x,y],z\right)+\omega\left([x,z],y\right)$ extends to a left...

We show that any left invariant metric with harmonic curvature on a solvable Lie group is Ricci-parallel. We show the same result for any Lie group of dimension ≤ 6.

Kundt spacetimes are of great importance to General Relativity. We show that a Kundt spacetime is a Lorentz manifold with a non-singular isotropic geodesic vector field having its orthogonal distribution integrable and determining a totally geodesic foliation. We give the local structure of Kundt spacetimes and some properties of left invariant Kun...

A Koszul–Vinberg manifold is a manifold M endowed with a pair \((\nabla ,h)\) where \(\nabla \) is a flat connection and h is a symmetric bivector field satisfying a generalized Codazzi equation. The geometry of such manifolds could be seen as a type of bridge between Poisson geometry and pseudo-Riemannian geometry, as has been highlighted in our p...

Let ( M , ∇, 〈 , 〉 ) be a manifold endowed with a flat torsionless connection r and a Riemannian metric 〈 , 〉 and ( T k M ) k ≥1 the sequence of tangent bundles given by T k M = T ( T k ⁻¹ M ) and T ¹ M = TM . We show that, for any k ≥ 1, T k M carries a Hermitian structure ( J k , g k ) and a flat torsionless connection ∇ k and when M is a Lie gro...

We classify symmetric Leibniz algebras in dimensions 3 and 4 and we determine all associated Lie racks. Some of such Lie racks give rise to nontrivial topological quandles. We study some algebraic properties of these quandles and we give a necessary and sufficient condition for them to be quasi-trivial.

We show that any left invariant metric with harmonic curvature on a solvable Lie group is Ricci-parallel. We show the same result for any Lie group of dimension $\leq$ 6.

Let (M, ∇, ,) be a manifold endowed with a flat torsionless connection ∇ and a Riemannian metric , and (T k M) k≥1 the sequence of tangent bundles given by T k M = T (T k−1 M) and T 1 M = T M. We show that, for any k ≥ 1, T k M carries a Hermitian structure (J k , g k) and a flat torsionless connection ∇ k and when M is a Lie group and (∇, ,) are l...

We give a complete classification of left invariant generalized complex structures of type 1 on four dimensional simply connected Lie groups and we compute for each class its invariant generalized Dolbeault cohomology, its invariant generalized Bott-Chern cohomology and its invariant generalized Aeppli cohomology. We classify also left invariant ge...

We classify biharmonic and harmonic homomorphisms f:(G,g1)⟶(G,g2) where G is a connected and simply connected three-dimensional unimodular Lie group and g1 and g2 are left invariant Riemannian metrics.

k-Para-Kähler Lie algebras are a generalization of para-Kähler Lie algebras (k = 1) and constitute a subclass of k-symplectic Lie algebras. In this paper, we show that the characterization of para-Kähler Lie algebras as left symmetric bialgebras can be generalized to k-para-Kähler Lie algebras leading to the introduction of two new structures which...

A Koszul-Vinberg manifold is a manifold $M$ endowed with a pair $(\nabla,h)$ where $\nabla$ is a flat connection and $h$ is a symmetric bivector field satisfying a generalized Codazzi equation. The geometry of such manifolds could be seen as a type of bridge between Poisson geometry and pseudo-Riemannian geometry, as has been highlighted in our pre...

Oscillator Lie algebras are the only non commutative solvable Lie algebras which carry a bi-invariant Lorentzian metric. In this paper, we determine all the Poisson structures, and in particular, all symmetric Leibniz algebra structures whose underlying Lie algebra is an oscillator Lie algebra. We give also all the symmetric Leibniz bialgebra struc...

Given a symmetric Leibniz algebra $(\mathcal{L},.)$, the product is Lie-admissible and defines a Lie algebra bracket $[\;,\;]$ on $\mathcal{L}$. Let $G$ be the connected and simply-connected Lie group associated to $(\mathcal{L},[\;,\;])$. We endow $G$ with a Lie rack structure such that the right Leibniz algebra induced on $T_eG$ is exactly $(\mat...

Let (M,⟨,⟩TM) be a Riemannian manifold. It is well known that the Sasaki metric on TM is very rigid, but it has nice properties when restricted to T(r)M={u∈TM,|u|=r}. In this paper, we consider a general situation where we replace TM by a vector bundle E⟶M endowed with a Euclidean product ⟨,⟩E and a connection ∇E which preserves ⟨,⟩E. We define the...

k$-Para-K\"ahler Lie algebras are a generalization of para-K\"ahler Lie algebras $(k=1)$ and constitute a subclass of $k$-symplectic Lie algebras. In this paper, we show that the characterization of para-K\"ahler Lie algebras as left symmetric bialgebras can be generalized to $k$-para-K\"ahler Lie algebras leading to the introduction of two new str...

We classify biharmonic and harmonic homomorphisms $f:(G,g_1)\rightarrow(G,g_2)$ where $G$ is a connected and simply connected three-dimensional unimodular Lie group and $g_1$ and $g_2$ are left invariant Riemannian metrics.

We give a complete classification of Einstein Lorentzian 3-nilpotent simply connected Lie groups with 1-dimensional nondegenerate center.

We give a complete classification of left invariant generalized complex structures of type 1 on four dimensional simply connected Lie groups and we compute for each class its invariant generalized Dolbeault cohomology, its invariant generalized Bott-Chern cohomology and its invariant generalized Aeppli cohomology. We classify also left invariant ge...

We give a complete classification of left invariant generalized complex structures of type 1 on four dimensional simply connected Lie groups and we compute for each class its invariant generalized Dolbeault cohomology, its invariant generalized Bott-Chern cohomology and its invariant generalized Aeppli cohomology. We classify also left invariant ge...

A contravariant pseudo-Hessian manifold is a manifold M endowed with a pair (∇,h) where ∇ is a flat connection and h is a symmetric bivector field satisfying a contravariant Codazzi equation. When h is invertible we recover the known notion of pseudo-Hessian manifold. Contravariant pseudo-Hessian manifolds have properties similar to Poisson manifol...

In this paper, we study Lorentzian left invariant Einstein metrics on nilpotent Lie groups. We show that if the center of such Lie groups is degenerate then they are Ricci-flat and their Lie algebras can be obtained by the double extension process from an abelian Euclidean Lie algebra. We show that all nilpotent Lie groups up to dimension 5 endowed...

Oscillator Lie algebras are the only non commutative solvable Lie algebras which carry a bi-invariant Lorentzian metric. In this paper, we determine all the Poisson structures, and in particular, all symmetric Leibniz algebra structures whose underlying Lie algebra is an oscillator Lie algebra. We give also all the symmetric Leibniz bialgebra struc...

A linear Lie rack structure on a finite dimensional vector space V is a Lie rack operation (x,y)↦x⊳y pointed at the origin and such that for any x, the left translation Lx:y↦Lx(y)=x⊳y is linear. A linear Lie rack operation ⊳ is called analytic if for any x,y∈V, x⊳y=y+∑n=1∞An,1(x,…,x,y), where An,1:V×⋯×V→V is an n + 1-multilinear map symmetric in th...

A contravariant pseudo-Hessian manifold is a manifold $M$ endowed with a pair $(\nabla,h)$ where $\nabla$ is a flat connection and $h$ is a symmetric bivector field satisfying a contravariant Codazzi equation. When $h$ is invertible we recover the known notion of pseudo-Hessian manifold. Contravariant pseudo-Hessian manifolds have properties simila...

In this paper, we study Lorentzian left invariant Einstein metrics on nilpotent Lie groups. We show that if the center of such Lie groups is degenerate then they are Ricci-flat and their Lie algebras can be obtained by the double extension process from an abelian Euclidean Lie algebra. We show that all nilpotent Lie groups up to dimension $5$ endow...

A Riemann-Poisson Lie group is a Lie group endowed with a left invariant Riemannian metric and a left invariant Poisson tensor which are compatible in the sense introduced in C.R. Acad. Sci. Paris s\'er. {\bf I 333} (2001) 763-768. We study these Lie groups and we give a characterization of their Lie algebras. We give also a way of building these L...

A linear Lie rack structure on a finite dimensional vector space $V$ is a Lie rack operation $(x,y)\mapsto x\rhd y$ pointed at the origin and such that for any $x$, the left translation $\mathrm{L}_x:y\mapsto \mathrm{L}_x(y)= x\rhd y$ is linear. A linear Lie rack operation $\rhd$ is called analytic if for any $x,y\in V$, \[ x\rhd y=y+\sum_{n=1}^\in...

In this paper, we generalize all the results obtained on para‐Kähler Lie algebras in [3] to para‐Kähler Lie algebroids. In particular, we study exact para‐Kähler Lie algebroids as a generalization of exact para‐Kähler Lie algebras. This study leads to a natural generalization of pseudo‐Hessian manifolds, we call them contravariant pseudo‐Hessian Ma...

There are five unimodular simply connected three dimensional unimodular non abelian Lie groups: the nilpotent Lie group $\mathrm{Nil}$, the special unitary group $\mathrm{SU}(2)$, the universal covering group $\widetilde{\mathrm{PSL}}(2,\mathbb{R})$ of the special linear group, the solvable Lie group $\mathrm{Sol}$ and the universal covering group...

There are five unimodular simply connected three dimensional unimodular non abelian Lie groups: the nilpotent Lie group Nil, the special unitary group SU(2), the universal covering group PSL(2, R) of the special linear group, the solvable Lie group Sol and the universal covering group E 0 (2) of the connected component of the Euclidean group. For e...

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 h p,q of natural metrics on the tangent bundle depending on two parameters with p∈R and q≥0. This family possesses interesting ge...

Let $(M,\langle,\rangle_{TM})$ be a Riemannian manifold. It is well-known that the Sasaki metric on $TM$ is very rigid but it has nice properties when restricted to $T^{(r)}M=\{u\in TM,|u|=r \}$. In this paper, we consider a general situation where we replace $TM$ by a vector bundle $E\longrightarrow M$ endowed with a Euclidean product $\langle,\ra...

Let (M, , T M) be a Riemannian manifold. It is well-known that the Sasaki metric on T M is very rigid but it has nice properties when restricted to T (r) M = {u ∈ T M, |u| = r}. In this paper, we consider a general situation where we replace T M by a vector bundle E −→ M endowed with a Euclidean product , E and a connection ∇ E which preserves , E....

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 re...

Let $M$ be a smooth manifold and $\Gamma$ a group acting on $M$ by diffeomorphisms; which means that there is a group morphism $\rho:\Gamma\rightarrow \mathrm{Diff}(M)$ from $\Gamma$ to the group of diffeomorphisms of $M$. For any such action we associate a cohomology $\mathrm{H}(\Omega(M)_\Gamma)$ which we call the cohomology of $\Gamma$-coinvaria...

A flat pseudo-Euclidean Lie algebra is a real Lie algebra with a non degenerate symmetric bilinear form and a left symmetric product whose the commutator is the Lie bracket and such that the left multiplications are skew-symmetric. We show that the center of a flat pseudo-Euclidean nilpotent Lie algebra of signature $(2,n-2)$ must be degenerate and...

In this paper, we study different aspects of harmonic and biharmonic homomorphisms between Riemannian Lie groups. From this class we develop methods to construct many new examples of biharmonic maps.

In this paper, we generalize all the results obtained on para-K\"ahler Lie algebras in Journal of Algebra {\bf 436} (2015) 61-101 to para-K\"ahler Lie algebroids. In particular, we study exact para-K\"ahler Lie algebroids as a generalization of exact para-K\"ahler Lie algebras. This study leads to a natural generalization of pseudo-Hessian manifold...

We give a complete description of semi-symmetric algebraic curvature tensors on a four-dimensional Lorentzian vector space and we use this description to determine all four-dimensional homogeneous semi-symmetric Lorentzian manifolds.

We give a complete description of semi-symmetric algebraic curvature tensors on a four-dimensional Lorentzian vector space and we use this description to determine all four-dimensional homogeneous semi-symmetric Lorentzian manifolds.

In this study, we consider Lie algebras that admit para-Kahler and hyper-para-Kahler structures. We provide new characterizations of these Lie algebras and develop many methods for building large classes of examples. Previously, Bai considered para-Kahler Lie algebras as left symmetric bialgebras. We reconsider this viewpoint and make improvements...

We study the signature of Ricci curvature of all left invariant Riemannian metrics on a given nilpotent Lie group. We determine such signatures for some classes of nilpotent Lie groups

Let $(G,h)$ be a nilpotent Lie group endowed with a left invariant Riemannian metric, $\mathfrak{g}$ its Euclidean Lie algebra and $Z(\mathfrak{g})$ the center of $\mathfrak{g}$. By using an orthonormal basis adapted to the splitting $\mathfrak{g}=(Z(\mathfrak{g})\cap[\mathfrak{g},\mathfrak{g}])\oplus O^+\oplus (Z(\mathfrak{g})\cap[\mathfrak{g},\ma...

A Lie group $G$ endowed with a left invariant Riemannian metric $g$ is called Riemannian Lie group. Harmonic and biharmonic maps between Riemannian manifolds is an important area of investigation. In this paper, we study different aspects of harmonic and biharmonic homomorphisms between Riemannian Lie groups. We show that this class of biharmonic m...

We study Lie algebras admitting para-Kähler and hyper-para-Kähler structures. We give new characterizations of these Lie algebras and we develop many methods to build large classes of examples. Bai considered para-Kähler Lie algebras as left symmetric bialgebras. We reconsider this point of view and improve it in order to obtain some new results. T...

A Lorentzian flat Lie group is a Lie group $G$ with a flat left invariant metric $\mu$ with signature $(1,n-1)=(-,+,\ldots,+)$. The Lie algebra $\mathfrak{g}=T_eG$ of $G$ endowed with $\mu(e)$ is called flat Lorentzian Lie algebra. It is known that the metric of a flat Lorentzian Lie group is geodesically complete if and only if its Lie algebra is...

Let $(M,\pi,\mathcal{D})$ be a Poisson manifold endowed with a flat, torsion-free contravariant connection. We show that if $\mathcal{D}$ is an $\mathcal{F}$-connection then there exists a tensor $\mathbf{T}$ such that $\mathcal{D}\mathbf{T}$ is the metacurvature tensor introduced by E. Hawkins in his work on noncommutative deformations. We compute...

Let $G$ be a connected Lie group and $\mathfrak{g}$ its Lie algebra. We denote by $\nabla^0$ the torsion free bi-invariant linear connection on $G$ given by $\nabla^0_XY=\frac12[X,Y],$ for any left invariant vector fields $X,Y$. A Poisson structure on $\mathfrak{g}$ is a commutative and associative product on $\mathfrak{g}$ for which $\mathrm{ad}_u...

We study Lie algebras admitting para-K\"ahler and hyper-para-K\"ahler structures. We give new characterizations of these Lie algebras and we develop many methods to build large classes of examples. Bai considered para-K\"ahler Lie algebras as left symmetric bialgebras. We reconsider this point of view and improve it in order to obtain some new resu...

We give a method to lift $(2,0)$-tensors fields on a manifold $M$ to build symplectic forms on $TM$. Conversely, we show that any symplectic form $\Om$ on $TM$ is symplectomorphic, in a neighborhood of the zero section, to a symplectic form built naturally from three $(2,0)$-tensor fields associated to $\Om$.

A Lie group is called quadratic if it carries a bi-invariant semi-Riemannian metric. Oscillator Lie groups constitute a subclass of the class of quadratic 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 algebras. Moreover, we show that...

A n-dimensional Lie group G equipped with a left invariant symplectic form ω + is called a symplectic Lie group. It is well-known that ω + induces a left invariant affine structure on G. Relatively to this affine structure we show that the left invariant Poisson tensor π + corresponding to ω + is polynomial of degree 1 and any right invariant k-mul...

We call the Lie algebra of a Lie group with a left invariant pseudo-Riemannian flat metric pseudo-Riemannian flat Lie algebra. We give a new proof of a classical result of Milnor on Riemannian flat Lie algebras. We reduce the study of Lorentzian flat Lie algebras to those with trivial center or those with degenerate center. We show that the double...

Let M be a smooth manifold endowed with a Poisson tensor σ and a Riemannian metric g and let J=σ#∘#−1 be the (1,1) tensor field relating σ to g. It is well known that the complete lift of J defines a bivector field ΠJ on T⁎M which is a Poisson tensor compatible with canonical Poisson structure on T⁎M if J is torsionless. We consider the Lie algebro...

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, w...

We determine all Ricci flat left invariant Lorentzian metrics on simply connected 2-step nilpotent Lie groups. We show that the $2k+1$-dimensional Heisenberg Lie group $H_{2k+1}$ carries a Ricci flat left invariant Lorentzian metric if and only if $k=1$. We show also that for any $2\leq q\leq k$, $H_{2k+1}$ carries a Ricci flat left invariant pseud...

We show that, on a compact symmetric space, the Lichnerowicz Laplacian acting on the space of covariant tensor fields coincides with the Casimir operator and we deduce that, on a compact semisimple Lie group, the Lichnerowicz Laplacian is the mean of the left invariant Casimir operator and the right invariant Casimir operator. To cite this article:...

Let Hn be the Heisenberg group of dimension 2n+1. We give a precise description of all (π,〈,〉), where π is a multiplicative Poisson tensor on Hn and 〈,〉 is a left invariant metric on Hn such that (π,〈,〉) satisfies the necessary conditions, introduced by Eli Hawkins, to the existence of a noncommutative deformation of the spectral triple associated...

We study the triple $(G,\pi,\prs)$ where $G$ is a connected and simply connected Lie group, $\pi$ and $\prs$ are, respectively, a multiplicative Poisson tensor and a left invariant Riemannian metric on $G$ such that the necessary conditions, introduced by Hawkins, to the existence of a non commutative deformation (in the direction of $\pi$) of the...

We study the triples $(G,\pi,〈,〉)$, where $G$ is a connected and simply connected Lie group, $\pi$ is a multiplicative Poisson tensor and $〈,〉$ is a left invariant Riemann metric such that the Hawkins conditions [E. Hawkins, Commun. Math. Phys. 246, No. 2, 211–235 (2004; Zbl 1055.58001); J. Differ. Geom. 77, No. 3, 385–424 (2007; Zbl 1130.53062)] a...

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.

A $n$-dimensional Lie group $G$ equipped with a left invariant symplectic form $\om^+$ is called a symplectic Lie group. It is well-known that $\om^+$ induces a left invariant affine structure on $G$. Relatively to this affine structure we show that the left invariant Poisson tensor $\pi^+$ corresponding to $\om^+$ is polynomial of degree 1 and any...

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.

We show that given a finite-dimensional real Lie algebra G{\mathcal{G}} acting on a smooth manifold P then, for any solution of the classical Yang–Baxter equation on G{\mathcal{G}} , there is a canonical Poisson tensor on P and an associated canonical torsion-free and flat contravariant connection. Moreover, we prove that the metacurvature of this...

We compute the eigenvalues with multiplicities of the Lichnerowicz Laplacian acting on the space of symmetric covariant tensor fields on the Euclidian sphere $S^n$. The spaces of symmetric eigentensors are explicitly given.

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}...

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.

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 autho...

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. A Riemannian interpretation of the Reeb class will give some geometric criteria...

Riemann-Poisson manifolds were introduced by the author in C. R. Acad. Sci. Paris, Ser. I 333 (2001) 763-768, and studied in detail in preprint math.DG/0206102. Kähler-Riemann foliations form an interesting subset of the Riemannian foliations with remarkable properties (see Ann. Global Anal. Geom. 21 (2002) 377-399). In this Note we will show that...

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 reg...

Riemann Poisson manifolds were introduced by the author in [1] and studied in more details in [2]. K\"ahler-Riemann foliations form an interesting subset of the Riemannian foliations with remarkable properties (see [3]). In this paper we will show that to give a regular Riemann Poisson structure on a manifold $M$ is equivalent to to give a K\"ahler...

In a previous paper (C. R. Acad. Sci. Paris Sér. I 333 (2001) 763–768), the author introduced a notion of compatibility between a Poisson structure and a pseudo-Riemannian metric. In this paper, we introduce a new class of Lie algebras called pseudo-Riemannian Lie algebras. The two notions are closely related: we prove that the dual of a Lie algebr...

We will introduce two notions of compatibility bettwen pseudo-Riemannian metric and Poisson structure using the notion of contravariant connection introduced by Fernandes R. L., we will study some proprities of manifold endowed with such compatible structures an we will give some examples.

We will introduce two notions of compatibility between pseudo-Riemannian metric and Poisson structure using the notion of contravariant connection introduced in [1], we will study some properties of manifold endowed with such compatible structures and we will give some examples.

In this paper, we compute the spectra of the Lichnerowicz Laplacian on the symmetric forms of degree 2 on CPn. This obtained by using the Riemannian submersion of S2n+1 to CPn and the computation done on S2n+1 in [6].

In this paper, we compute the spectrum of the Lichnerowicz laplacian on the symmetric forms of degree 2 on the sphere $S^n$ and the real projective space ${\mathbb R} P^n$. This is obtained by generalizing to forms the calculations of the spectrum of the laplacian on fonctions done via restriction of harmonic polynomials on euclidean space.

In this paper, we compute the spectrum of the Lichnerowicz laplacian on the symmetric forms of degree 2 on the sphere $S^n$ and the real projective space ${\mathbb R} P^n$. This is obtained by generalizing to forms the calculations of the spectrum of the laplacian on fonctions done via restriction of harmonic polynomials on euclidean space.

Soit (M, g) une variété riemannienne. Nous montrons que l'espace vectoriel G des formes symétriques invariantes par le flot géodésique est une algèbre de Lie contenant (comme sous-algèbre) l'algèbre des champs de Killing ainsi que l'espace vectoriel des formes symétriques parallèles comme sous-algèbre abélienne. Dans un deuxième temps, nous donnons...

Let (M, g) be a Riemannian manifold. We prove that the space of symmetric tensors invariant under the geodesic flow, is a Lie algebra which contains, as a subalgebra, the Lie algebra of Killing vector fields, and which also contains the space of parallel symmetric tensors as an Abelian subalgebra. Morever, we give a Weitzenböck decomposition of som...

Toutes les structures considérées dans ce travail sont de classe C ∞. Précisons tout d’abord la notion de complète intégrabilité utilisée ici. Rappelons qu’un système hamiltonien (M 2 n , ω, H) est dit complètement intégrable au sens d’Arnold-Liouville s’il existe un n-uple F = (f 1,..., f n )d’intégrales premières en involution dont les différenti...