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A pseudo-Euclidean non-associative algebra $(\mathfrak{g}, \bullet)$ is a real algebra of finite dimension that has a metric, i.e., a bilinear, symmetric, and non-degenerate form $\langle\;\rangle$. The metric is considered $\mathrm{L}$-invariant (resp. $\mathrm{R}$-invariant) if all left multiplications (resp. right multiplications) are skew-symmetric. The metric is called associative if $\langle u\bullet v,w\rangle= \langle u,v\bullet w\rangle$ for all $u, v, w \in \mathfrak{g}$. These three notions coincide when $\mathfrak{g}$ is a Lie algebra and in this case $\mathfrak{g}$ endowed with the metric is known as a quadratic Lie algebra. This paper provides a complete description of $\mathrm{L}$-invariant, $\mathrm{R}$-invariant, or associative pseudo-Euclidean metrics on left Leibniz algebras. It shows that a left Leibniz algebra with an associative metric is also right Leibniz and can be obtained easily from its underlying Lie algebra, which is a quadratic Lie algebra. Additionally, it shows that at the core of a left Leibniz algebra endowed with a $\mathrm{L}$-invariant or $\mathrm{R}$-invariant metric, there are two Lie algebras with one quadratic and the left Leibniz algebra can be built from these Lie algebras. We derive many important results from these complete description. Finally, the paper provides a list of left Leibniz algebras with an associative metric up to dimension 6, as well as a list of left Leibniz algebras with an $\mathrm{L}$-invariant metric, up to dimension 4, and $\mathrm{R}$-invariant metric up to dimension 5., 30 pages

$k$-symplectic manifolds are a convenient framework to study classical field theories and they are a generalization of polarized symplectic manifolds. This paper focus on the existence and the properties of left invariant $k$-symplectic structures on Lie groups having a bi-invariant pseudo-Riemannian metric. We show that compact semi-simple Lie groups and a large class of Lie groups having a bi-invariant pseudo-Riemannian metric does not carry any left invariant $k$-symplectic structure. This class contains the oscillator Lie groups which are the only solvable non abelian Lie groups having a bi-invariant Lorentzian metric. However, we built a natural left invariant $n$-symplectic structure on $\mathrm{SL}(n,\mathbb{R})$. Moreover, up to dimension 6, only three connected and simply connected Lie groups have a bi-invariant indecomposable pseudo-Riemannian metric and a left invariant k-symplectic structure, namely, the universal covering of $\mathrm{SL}(2, \mathbb{R})$ with a 2-symplectic structure, the universal covering of the Lorentz group $\mathrm{SO}(3, 1)$ with a 2-symplectic structure, and a 2-step nilpotent 6-dimensional connected and simply connected Lie group with both a 1-symplectic structure and a 2-symplectic structure., 16 pages

Let $(M,\nabla,\langle\;,\;\rangle)$ be a manifold endowed with a flat torsionless connection $\nabla$ and a Riemannian metric $\langle\;,\;\rangle$ and $(T^kM)_{k\geq1}$ the sequence of tangent bundles given by $T^kM=T(T^{k-1}M)$ and $T^1M=TM$. We show that, for any $k\geq1$, $T^kM$ carries a Hermitian structure $(J_k,g_k)$ and a flat torsionless connection $\nabla^k$ and when $M$ is a Lie group and $(\nabla,\langle\;,\;\rangle)$ are left invariant there is a Lie group structure on each $T^kM$ such that $(J_k,g_k,\nabla^k)$ are left invariant. It is well-known that $(TM,J_1,g_1)$ is K\"ahler if and only if $\langle\;,\;\rangle$ is Hessian, i.e, in each system of affine coordinates $(x_1,\ldots,x_n)$, $\langle\partial_{x_i},\partial_{x_j}\rangle=\frac{\partial^2\phi}{\partial_{x_i}\partial_{x_j}}$. Having in mind many generalizations of the K\"ahler condition introduced recently, we give the conditions on $(\nabla,\langle\;,\;\rangle)$ so that $(TM,J_1,g_1)$ is balanced, locally conformally balanced, locally conformally K\"ahler, pluriclosed, Gauduchon, Vaismann or Calabi-Yau with torsion. Moreover, we can control at the level of $(\nabla,\langle\;,\;\rangle)$ the conditions insuring that some $(T^kM,J_k,g_k)$ or all of them satisfy a generalized K\"ahler condition. For instance, we show that there are some classes of $(M,\nabla,\langle\;,\;\rangle)$ such that, for any $k\geq1$, $(T^kM,J_k,g_k)$ is balanced non-K\"ahler and Calabi-Yau with torsion. By carefully studying the geometry of $(M,\nabla,\langle\;,\;\rangle)$, we develop a powerful machinery to build a large classes of generalized K\"ahler manifolds., Comment: 40 pages, 8 Tables

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 Kundt structures on Lie groups. Finally, we classify all left invariant Kundt structures on three dimensional simply connected unimodular Lie groups., Comment: 15 pages

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 previous article [\textit{Contravariant Pseudo-Hessian manifolds and their associated Poisson structures}. \rm{Differential Geometry and its Applications} (2020)]. Our objective here will be to pursue our study by focusing in this setting on submanifolds by taking into account some developments in the theory of Poisson submanifolds., 27 pages

$k$-Para-K��hler Lie algebras are a generalization of para-K��hler Lie algebras $(k=1)$ and constitute a subclass of $k$-symplectic Lie algebras. In this paper, we show that the characterization of para-K��hler Lie algebras as left symmetric bialgebras can be generalized to $k$-para-K��hler Lie algebras leading to the introduction of two new structures which are different but both generalize the notion of left symmetric algebra. This permits also the introduction of generalized $S$-matrices. We determine then all the $k$-symplectic Lie algebras of dimension $(k+1)$ and all the six dimensional 2-para-K��hler Lie algebras., 21 pages, 3 Tables

We give a complete classification of Einstein Lorentzian 3-nilpotent simply connected Lie groups with 1-dimensional nondegenerate center., Comment: 21 pages submitted

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 $\widetilde{\mathrm{E}_0}(2)$ of the connected component of the Euclidean group. For each $G$ among these Lie groups, we give explicitly the list of all Lorentzian left invariant metrics on $G$, up to un automorphism of $G$. Moreover, for any Lorentzian left invariant metric in this list we give its Ricci curvature, scalar curvature, the signature of the Ricci curvature and we exhibit some special features of these curvatures. Namely, we give all the metrics with constant curvature, semi-symmetric non locally symmetric metrics and the Ricci solitons., Comment: 10 pages submitted

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 $\Gamma$-coinvariant forms. This is the cohomology of the graded vector space generated by the differentiable forms $\omega -\rho(\gamma)^*\omega$ where $\omega$ is a differential form with compact support and $\gamma\in \Gamma$. The present paper is an introduction to the study of this cohomology. More precisely, we study the relations between this cohomology, the de Rham cohomology and the cohomology of invariant forms $\mathrm{H}(\Omega(M)^\Gamma)$ in the case of isometric actions on compact Riemannian oriented manifolds and in the case of properly discontinuous actions on manifolds., Comment: To appear in Journal of Lie theory

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 manifolds. Generalized pseudo-Hessian manifolds have many similarities with Poisson manifolds. We explore these similarities which, among others, leads to a powerful machinery to build examples of non trivial pseudo-Hessian structures. Namely, we will show that given a finite dimensional commutative and associative algebra $(\mathcal{A},.)$, the orbits of the action $\Phi$ of $(\mathcal{A},+)$ on $\mathcal{A}^*$ given by $\Phi(a,\mu)=\exp(L_a^*)(\mu)$ are pseudo-Hessian manifolds, where $L_a(b)=a.b$. We illustrate this result by considering many examples of associative commutative algebras an show that the pseudo-Hessian manifolds obtained are very interesting., Comment: 23 pages

In this paper, we generalize all the results obtained on para-K��hler Lie algebras in Journal of Algebra {\bf 436} (2015) 61-101 to para-K��hler Lie algebroids. In particular, we study exact para-K��hler Lie algebroids as a generalization of exact para-K��hler Lie algebras. This study leads to a natural generalization of pseudo-Hessian manifolds. Generalized pseudo-Hessian manifolds have many similarities with Poisson manifolds. We explore these similarities which, among others, leads to a powerful machinery to build examples of non trivial pseudo-Hessian structures. Namely, we will show that given a finite dimensional commutative and associative algebra $(\mathcal{A},.)$, the orbits of the action $��$ of $(\mathcal{A},+)$ on $\mathcal{A}^*$ given by $��(a,��)=\exp(L_a^*)(��)$ are pseudo-Hessian manifolds, where $L_a(b)=a.b$. We illustrate this result by considering many examples of associative commutative algebras an show that the pseudo-Hessian manifolds obtained are very interesting., 23 pages

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 maps can be used at the first level to build examples but, as we will see through this paper, its study will lead to some interesting mathematical problems in the theory of Riemannian Lie groups., Some new results are added

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 $\langle\;,\;\rangle=\mu(e)$ 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 unimodular. In this paper, we characterise nonunimodular Lorentzian flat Lie algebras as double extensions (in the sense of Aubert-Medina \cite{Aub-Med}) of Riemannian flat Lie algebras. As application of this result, we give all nonunimodular Lorentzian flat Lie algebras up to dimension 4., Comment: 12 pages some modifications of the first version

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 results. The study of para-K\"ahler and hyper-para-K\"ahler is intimately linked to the study of left symmetric algebras and, in particular, those admitting invariant symplectic forms. In this paper, we give many new classes of left symmetric algebras and a complete description of all associative algebras admitting an invariant symplectic form. We give also all four dimensional hyper-para-K\"ahler Lie algebras.

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 $\Om$., Submitted to C.R.A.S

We study Lie algebras admitting para-K��hler and hyper-para-K��hler structures. We give new characterizations of these Lie algebras and we develop many methods to build large classes of examples. Bai considered para-K��hler Lie algebras as left symmetric bialgebras. We reconsider this point of view and improve it in order to obtain some new results. The study of para-K��hler and hyper-para-K��hler is intimately linked to the study of left symmetric algebras and, in particular, those admitting invariant symplectic forms. In this paper, we give many new classes of left symmetric algebras and a complete description of all associative algebras admitting an invariant symplectic form. We give also all four dimensional hyper-para-K��hler Lie algebras.

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 $\mathrm{ad}_u$ is a derivation, for any $u\in\mathfrak{g}$. A torsion free bi-invariant linear connections on $G$ which have the same curvature as $\nabla^0$ is called special. We show that there is a bijection between the space of special connections on $G$ and the space of Poisson structures on $\mathfrak{g}$. We compute the holonomy Lie algebra of a special connection and we show that the Poisson structures associated to special connections which have the same holonomy Lie algebra as $\nabla^0$ possess interesting properties. Finally, we study Poisson structures on a Lie algebra and we give a large class of examples which gives, of course, a large class of special connections., 31 pages, This research was conducted within the framework of Action concert\'ee CNRST-CNRS Project SPM04/13

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 spectrale triple associated to $\prs$ are satisfied. We show that the geometric problem of the classification of such triple $(G,\pi,\prs)$ is equivalent to an algebraic one. We solve this algebraic problem in low dimensions and we give the list of all $(G,\pi,\prs)$ satisfying Hawkins's conditions, up to dimension four., Comment: 23 pages

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 pseudo-Riemannian metric of signature $(q,2k+1-q)$ and we give explicite examples of such metrics., Comment: Title modified, some explicites examples are given and Section 6 deleted

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.

Multiple myeloma is a hematopoetic disorder and multicentric disease, with the most common localisation being the spine. A 47-year-old male presented with progressive paraplegia, superficial and deep sensory disturbance below the level of T4. Spinal magnetic resonance image showed an epidural mass compressing the spinal cord at the level of T4-T6 with intact bone structure. The patient underwent surgical T4-T6 posterior spinal decompression. Microscopic examination and immunohistochemical studies confirmed the diagnosis of multiple myeloma of kappa subtype. Immunoelectrophoresis revealed the presence of immunoglobulin G kappa. The patient was subsequently started on steroids and chemotherapy for myeloma. Extraosseous epidural tumors causing compression myelopathy without evidence of destruction or collapse of vertebral bodies are relatively rare; to our knowledge only four cases have been reported in English literature.

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 pages

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 pages

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-Riemann foliation on $M$ such that the leafwise symplectic form is invariant with respect to all local foliate perpendicular vector fields. We show also that the sum of the vector space of leafwise cohomology and the vector space of basic forms is a subspace of the space of Poisson cohomology., Comment: 12 pages

Riemann Poisson manifolds were introduced by the author in [1] and studied in more details in [2]. K��hler-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��hler-Riemann foliation on $M$ such that the leafwise symplectic form is invariant with respect to all local foliate perpendicular vector fields. We show also that the sum of the vector space of leafwise cohomology and the vector space of basic forms is a subspace of the space of Poisson cohomology., 12 pages

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., Comment: 8 pages, to appear in C.R.A.S