Mohamed Boucetta, Oumaima Tibssirte
We give a complete classification of Einstein Lorentzian 3-nilpotent simply connected Lie groups with 1-dimensional nondegenerate center.
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Canonical Connections and Algebraic Ricci Solitons on Lorentzian 4-Dimensional Nilpotent Lie Groups
本文对带乘积结构的洛伦兹四维幂零李群，计算两类联络及其曲率并分类其对应的代数里奇孤子。
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 with a Lorentzian Einstein left invariant metric have degenerate center and we use this fact to give a complete classification of these metrics. We show that if $\mathfrak{g}$ is the Lie algebra of a nilpotent Lie group endowed with a Lorentzian l…
We present structural properties of Lie algebras admitting symmetric, invariant and nondegenerate bilinear forms. We show that these properties are not satisfied by nilradicals of parabolic subalgebras of real split forms of complex simple Lie algebras, neither by 2-step nilpotent Lie algebras associated with graphs, with only few exceptions. These rare cases are, essentially, abelian Lie algebras, the free 3-step nilpotent Lie algebra on 2-generators and the free 2-step nilpotent Lie algebra on 3-generators.
Conjugate loci of pseudoriemannian 2-step nilpotent Lie groups with nondegenerate center
我们确定了沿所有平行于或垂直于中心的测地线的完全共轭轨迹（定理2.3）。当中心是1维时，我们得到所有情况下的公式（定理2.5），当某个算子也是可对角化的时，这些公式就完全显式了（推论2.7）。这些结果给出了伪黎曼共轭轨迹光滑性的一些新的信息。我们还得到了所有共轭点的重数。
Curvature tensors whose Jacobi or Szabo operator is nilpotent on null vectors
证明了任意$k$ Osserman Lorentz代数曲率张量具有常截面曲率，并给出了任意局部2点齐次Lorentz流形具有常截面曲率的初等证明.我们还证明了Szabó Lorentzian的共变导数代数曲率张量为零。
It has been known that there exist exactly three left-invariant Lorentzian metrics up to scaling and automorphisms on the three dimensional Heisenberg group. In this paper, we classify left-invariant Lorentzian metrics on the direct product of three dimensional Heisenberg group and the Euclidean space of dimension $n-3$ with $n \geq 4$, and prove that there exist exactly six such metrics on this Lie group up to scaling and automorphisms. Moreover we show that only one of them is flat, and the other five metrics are Ricci solitons but not Einstein. We also characterize this flat metric as the u…
On the structure of $2$-step nilpotent Lorentzian naturally reductive Lie groups
研究关于特定表示群、具有自然约化左不变洛伦兹度量的两步幂零李群，通过弱化条件构建框架，涵盖中心非退化和退化情况，得到李代数、表示分解及等距自同构群相关结果，完成结构描述。
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.
Nonholonomic Lorentzian geometry on some $\mathbb H$-type groups
我们考虑的$\mathbb H$型群的自然水平分布所产生的组的对易关系的例子。与以往的研究不同，我们用指数为1的非退化Lorentzian度规代替了正定二次型来描述水平分布。因果特征被定义。研究了类时未来有向曲线的可达集。得到了测地线的参数方程。
New examples of $2$-nondegenerate real hypersurfaces in $\mathbb{C}^N$ with arbitrary nilpotent symbols
本文引入了一类在 $\mathbb{C}^N$（$N>3$）中具有秩 $1$ Levi 核的一致 $2$-非退化 CR 超曲面，构成了首个显式无限维族，解决了单 Jordan 块符号结构的等价问题与对称性分类，并证明多块符号结构可由单块结构构造。
The distribution of symmetry of Lorentzian naturally reductive nilmanifolds
本文研究了相对于某类等距传递子群自然约化的2步幂零洛伦兹李群，描述其迷向表示并证明其不动点产生对称性分布，推广了黎曼情形的已知结果。
Genericity of nondegenerate critical points and Morse geodesic functionals
考虑一类由巴拿赫流形的开子集参数化的Hilbert流形上的变分问题，并讨论了临界点的非退化条件的一般性.基于B.白色，我们证明了一个抽象的一般性结果，采用无限维的Sard-Smale定理。通过证明一般（非紧）半黎曼流形、正交分裂半黎曼流形和整体双曲Lorentz流形中不带退化测地线的度量的普适性，给出了应用.我们还讨论了定常洛伦兹流形中的通有性。
Let E be the Engel group and D be a rank 2 bracket generating left invariant distribution with a Lorentzian metric, which is a nondegenerate metric of index 1. In this paper, we first prove that timelike normal extremals are locally maximizing. Second, we obtain a parametrization of timelike, spacelike, lightlike normal extremal trajectories by Jacobi functions. Third, a discrete symmetry group and its fixed points which are Maxwell points of of timelike and spacelike normal extremals, are described. An estimate for the cut time (the time of loss of optimality) on extremal trajectories is deri…
The goal of this paper is to show that many key results found in the study of Einstein Lorentzian nilpotent Lie algebras can still hold in the more general settings of unimodular Lie algebras and (completely) solvable Lie algebras.
An absolute parallelism for $2$-nondegenerate CR manifolds $M$ of hypersurface type was recently constructed independently by Isaev-Zaitsev, Medori-Spiro, and Pocchiola in the minimal possible dimension ($\dim M=5$), and for $\dim M=7$ in certain cases by the first author. We develop a bigraded analog of Tanaka's prolongation procedure to construct a canonical absolute parallelism for these CR structures in arbitrary (odd) dimension with Levi kernel of arbitrary admissible dimension. We introduce the notion of a bigraded Tanaka symbol. Under regularity assumption that the symbol is a Lie algeb…
We define a class of Riemannian and pseudo-Riemannian 2-step nilpotent Lie groups with nondegenerate centers that generalize the H-type groups of Kaplan. Examples are given and geometric properties are investigated.
The goal of this paper is to investigate which one of thenon-isometric left invariant Lorentz metrics on 4-dimensional nilpotent Lie groups $H_3 \times {\Bbb R}$ and $G_4$ satisfy in Ricci Soliton equation. Among the left-invariant Lorentzian metrics on $H_3 \times {\Bbb R}$, ~ $g_λ^{+}$ is a shrinking while $g_λ^{+}$ and $g_μ$ are expanding and also $g_0^1, g_0^2, g_0^3$ have Ricci solitons. We exhibit among the non-isometric left invariant Lorentz metric on the group $G_4$ only $g_1^λ, g_2^λ$ have Lorentz Ricci solitons and $g_2^λ$ is a shrinking.
This paper deals with naturally reductive pseudo-Riemannian 2-step nilpotent Lie groups $(N, \la \,,\,\ra_N)$, such that $\la \,,\,\ra_N$ is invariant under a left action. The case of nondegenerate center is completely characterized. In fact, whenever $\la \,,\, \ra_N$ restricts to a metric in the center it is proved here that the simply connected Lie group $N$ can be constructed starting from a real representation $(π,\vv)$ of a certain Lie algebra $\ggo$. We study the geometry of $(N, \la \,,\,\ra_N)$ and we find the corresponding naturally reductive homogeneous structure. On the other hand,…
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 all flat pseudo-Euclidean nilpotent Lie algebras of signature $(2,n-2)$ can be obtained by using the double extension process from flat Lorentzian nilpotent Lie algebras. We show also that the center of a flat pseudo-Euclidean 2-step nilpotent Lie a…
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 Lorentz…
The aim of this work is the study of left-invariant magnetic fields on 2-step nilpotent Lie groups. While the existence of closed 2-forms for which the center is either nondegenerate or in the kernel of the 2-form, is always guaranteed, the existence of closed 2-forms for which the center is isotropic but not in the kernel of the 2-form, is a special situation. These 2-forms are called of type II. We obtain a strong obstruction for the existence on non-singular Lie algebras. Moreover, we prove that the only $H$-type Lie groups admitting closed 2-forms of type II are the real, complex and quate…
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 extension process can be used to construct all Lorentzian flat Lie algebras with degenerate center generalizing a result of Aubert-Medina on Lorentzian flat nilpotent Lie algebras. Finally, we give the list of Lorentzian flat Lie algebras with degene…
To determine the Lie groups that admit a flat (eventually complete) left invariant semi-Riemannian metric is an open and difficult problem. The main aim of this paper is the study of the flatness of left invariant semi Riemannian metrics on quadratic Lie groups i.e. Lie groups endowed with a bi-invariant semi Riemannian metric. We give a useful necessary and sufficient condition that guaranties the flatness of a left invariant semi Riemannian metric defined on a quadratic Lie group. All these semi Riemannian metrics are complete. We show that there are no Riemannian or Lorentzian flat left inv…
Harmful structures and Killing spinors on unimodular Lie groups
本文研究单模李群上的有害结构与 Killing 旋子，刻画其左不变有害结构，分类低维伪黎曼李群上的此类结构，相关李群对应特定爱因斯坦度量的主轨道。
Sub-Riemannian and sub-Lorentzian geodesics of the oscillator groups
本文研究振子群的次黎曼与次洛伦兹测地线，推广至半单李群与紧子群对应的Osc(G,K)，明确其测地线公式并实现为斯蒂弗尔丛。
We examine the existence of one parameter groups of diffeomorphisms whose infinitesimal generators annihilate all scalar polynomial curvature invariants through the application of the Lie derivative, known as $\mathcal{I}$-preserving diffeomorphisms. Such mappings are a generalization of isometries and appear to be related to nil-Killing vector fields, for which the associated Lie derivative of the metric yields a nilpotent rank two tensor. We show that the set of nil-Killing vector fields contains Lie algebras, although the Lie algebras may be infinite and can contain elements which are not $…
Double extension of flat pseudo-Riemannian $F$-Lie algebras
该研究定义平坦伪黎曼$F$-李代数并构造其双重扩张，证明其可生成特定类型李代数，还用于构造兼容平坦标量积的泊松代数，为相关代数结构研究提供新框架与方法。
$k$-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 b…
Cartan-Schouten metrics for information geometry and machine learning
本文研究Cartan-Schouten度量并将其作为信息几何模型，基于李群双不变均值提出一种用于机器学习的新参数化均值，并给出了该度量的显式恢复公式与完备几何刻画。
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