A. Gordillo, J. Navarro, P. Sancho
We present a simple remark that assures that the invariant theory of certain real Lie groups coincides with that of the underlying affine, real algebraic groups. In particular, this result applies to the non-compact orthogonal or symplectic Lie groups.
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We study local Lie algebras of pairs of functions which generate infinitesimal symmetries of almost-cosymplectic-contact structures of odd dimensional manifolds.
A remark about the Lie algebra of infinitesimal conformal transformations of the Euclidian space
$R^n$的无穷小共形变换总是多项式的，当$n>2$ .本文证明了$R^n$上无穷小共形多项式变换的李代数，$n>1$，在多项式向量场的李代数中是极大的.当$n$大于2且$p,q$使得$p+q=n$时，这意味着将$so(p+1,q+1,R)$嵌入到多项式向量场中的最大性，这在最近关于等变量化的工作中被重新讨论。它也改进了V.I.奥吉韦茨基
We use the notion of the principal three-dimensional subgroup of a simple Lie group to identify certain special subspaces of the Lie algebra and address the question of whether these are calibrated for invariant forms on the group.
We prove that any real Lie group of dimension \leq 5 admits a left invariant flat projective structure. We also prove that a real Lie group L of dimension \leq 5 admits a left invariant flat affine structure if and only if the Lie algebra of L is not perfect.
A remark on left invariant metrics on compact Lie groups
证明了紧李群$G$上的一个左不变度量在$\g$的一个子代数$\h$的方向上拉伸所得到的度量总是具有某些负的截面曲率，除非$\h$的半单部分是$\g$的理想.
We prove that "generalized Lie algebroid", a geometric object which appeared recently in the literature, is a misconception.
We consider deformations of left-invariant complex structures on simply connected semisimple compact Lie groups which are a priori non-invariant. Computing their cohomologies, we show that they are not actually biholomorphic to left-invariant (right-invariant) complex structures.
Invariant totally geodesic unit vector fields on three-dimensional Lie groups
给出了三维李群上具有左不变度量的左不变单位向量场的完整列表，这些左不变场在具有Sasaki度量的群的单位切丛中生成全测地子流形.结果，每一类三维李群都存在全测地单位向量场。从几何观点看，该场是群上的平行场或自然几乎切触结构的特征向量场。
Expository observation on the $μ$-invariant of singularity models for Ricci Flow.
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.
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 generalized Kähler structures on four dimensional simply connected Lie groups.
In this note, we completely classify left-invariant Riemann solitons on three-dimensional Lorentzian Lie groups.
Almost Kähler 4-dimensional Lie groups with $J$-invariant Ricci tensor
本文的目的是确定4维李群$(g, J, Ω)$上严格左不变的几乎Kähler结构，使得Ricci张量是$J$不变的。
We call a connected Lie group endowed with a left-invariant Lorentzian flat metric Lorentzian flat Lie group. In this Note, we determine all Lorentzian flat Lie groups admitting a timelike left-invariant Killing vector field. We show that these Lie groups are 2-solvable and unimodular and hence geodesically complete. Moreover, we show that a Lorentzian flat Lie group $(\mathrm{G},μ)$ admits a timelike left-invariant Killing vector field if and only if $\mathrm{G}$ admits a left-invariant Riemannian metric which has the same Levi-Civita connection of $μ$. Finally, we give an useful characteriza…
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…
In this paper, we classify Left-invariant Ricci collineations associated to Yano connections on three-dimensional Lorentzian Lie groups.
In this paper, we classify Left-invariant Ricci collineations associated to canonical connections and Kobayashi-Nomizu connections on three-dimensional Lorentzian Lie groups.
Remark on some conformally invariant integral equations: the method of moving spheres
利用移动球方法研究了一类共形不变的积分方程。
Foliations invariant under Lie group transverse actions
本文研究了在李群的横截作用下不变的（光滑的和全纯的）叶理。
$k$-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…
In this note, we consider a sequence of test configurations compatible with a Kaehler metric in $c_1(L)$ on a polarized algebraic manifold $(X,L)$. Then an explicit formula for the Donaldson-Futaki invariant $F_1$ for the sequence will be given.
Classification of left invariant Riemannian metrics on nonunimodular 4-dimensional Lie groups
本文通过自动机分类，研究了非单模4维连通李群上的左不变黎曼度量，等价于对非单模4维李代数的内积进行分类。
We find the index of symmetry for all solvable three-dimensional Lie groups with a left-invariant metric. When combined with the work of Reggiani on unimodular three-dimensional Lie groups, the index of symmetry is then known for all three-dimensional Lie groups with a left-invariant metric. Similar to the unimodular case, for every solvable three-dimensional Lie algebra there is at least one left-invariant metric for which the index of symmetry is positive. Also, the index is never equal to 2.
In the presented paper left-invariant pseudo-Riemannian metrics on four-dimensional Lie groups with zero Schouten-Weyl tensor are investigated. The complete classification of these metric Lie groups is obtained in terms of the structure constants of corresponding Lie algebras.
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.
We describe the full group of isometries of absolutely simple, compact, connected real Lie groups, of SO(4) and of U(n), endowed with suitable bi-invariant Riemannian metrics.
In this paper, we determine all left-invariant Ricci collineations associated to the Bott connection with three distributions on three-dimensional Lorentzian Lie groups . he author was supported in part by NSFC No.11771070.
Left-invariant harmonic spinors on three-dimensional Lie groups
研究三维李群上左不变调和旋量的存在性，通过修正并专门化旋量狄拉克算子公式，探讨李群结构方程的约束条件及对应度量的识别。
Left-invariant Codazzi tensors and harmonic curvature on Lie groups endowed with a left invariant Lorentzian metric
本文研究洛伦兹李群上的左不变 Codazzi 张量，分类 Ricci 算子的四种类型，并完整刻画具有调和曲率且 Ricci 算子非对角的洛伦兹李群李代数结构。
We find three-dimensional subspaces of four-dimensional connected Lie algebras, generating these algebras, and abnormal extremals on connected Lie groups with these Lie algebras and with left-invariant sub-Finsler quasimetrics defined by seminorms on such subspaces. In terms of the structure constants of Lie algebras and dual seminorms, we establish a criterion for the strong abnormality of these extremals.
We consider Lie groups that are either Heintze groups or Sol-type groups, which generalize the three-dimensional Lie group SOL. We prove that all left-invariant Riemannian metrics on each such a Lie group are roughly similar via the identity. This allows us to reformulate in a common framework former results by Le Donne-Xie, Eskin-Fisher-Whyte, Carrasco Piaggio, and recent results of Ferragut and Kleiner-Müller-Xie, on quasiisometries of these solvable groups.
We are interested in the class, in the Elie Cartan sense, of left invariant forms on a Lie group. We construct the class of Lie algebras provided with a contact form and classify the frobeniusian Lie algebras up to a contraction. We also study forms which are invariant by a subgroup. We show that the simple group SL(2n,R) which doesn't admit left invariant contact form, yet admits a contact form which is invariant by a maximal compact subgroup. We determine also Pfaffian forms on the Heisenberg $3$-dimensional group invariant by a subgroup and obtain the Transport Equation.
In this paper, we formulate a procedure to obtain a generalization of Milnor frames for left-invariant pseudo-Riemannian metrics on a given Lie group. This procedure is an analogue of the recent studies on left-invariant Riemannian metrics, and is based on the moduli space of left-invariant pseudo-Riemannian metrics. As one of applications, we show that any left-invariant pseudo-Riemannian metrics of arbitrary signature on the Lie groups of real hyperbolic spaces have constant sectional curvatures.
In this paper, we present the classification of all possible signatures of the Ricci curvature of left-invariant Riemannian metrics on 4-dimensional Lie groups and discuss some related questions.
Abnormal extremals on four-dimensional connected Lie groups with left-invariant sub-Finsler quasimetric, defined by a seminorm on a two-dimensional subspace of the Lie algebra generating the algebra, are found. In terms of structure constant of Lie algebra and supporting Minkowski function for the unit ball of seminorm on two-dimensional subspace of Lie algebra, defining a quasimetric, we establish a criterion for strict abnormality of these extremals.
Invariants of Lie algebroids
近年来，李代数胚的几个新的不变量被发现。我们给出了这些不变量的概述，并建立它们之间的几个关系。
Invariant and non-invariant almost complex structures on compact quotients of Lie groups
本文综述李代数复结构可容许性的经典问题，聚焦李群紧商空间上不变与非不变近复结构的行为差异，关注典范丛和小平维数，并提供了二者小平维数计算的新例子。
In this paper, we shall use a method based on the theory of extensions of left-symmetric algebras to classify complete left-invariant affine real structures on solvable non-unimodular three-dimensional Lie groups.
Gromov asked if the bi-invariant metrics on a compact Lie group are extremal compared to any other metrics. In this note, we prove that the bi-invariant metrics on a compact connected semi-simple Lie group $G$ are extremal (in fact rigid) in the sense of Gromov when compared to the left-invariant metrics. In fact the same result holds for a compact connected homogeneous manifold $G/H$ with $G$ compact connect and semi-simple.
For all left-invariant Riemannian metrics on three-dimensional unimodular Lie groups, there exist particular left-invariant orthonormal frames, so-called Milnor frames. In this paper, for any left-invariant Riemannian metrics on any Lie groups, we give a procedure to obtain an analogous of Milnor frames, in the sense that the bracket relations among them can be written with relatively smaller number of parameters. Our procedure is based on the moduli space of left-invariant Riemannian metrics. Some explicit examples of such frames and applications will also be given.
We investigate the infinitesimal invariants of an immersed submanifold $Σ$ of a Klein geometry $M\cong G/H$, and in particular an invariant filtration of Lie algebroids over $Σ$. The invariants are derived from the logarithmic derivative of the immersion of $Σ$ into $M$, a complete invariant introduced in the companion article, 'A characterization of smooth maps into a homogeneous space'. Applications of the Lie algebroid approach to subgeometry include a new interpretation of Cartan's method of moving frames and a novel proof of the fundamental theorem of hypersurfaces in Euclidean, elliptic …
We study left invariant conformal Killing-Yano (CKY) $2$-forms on Lie groups endowed with a left invariant metric. We classify all $5$-dimensional metric Lie algebras carrying CKY tensors that are obtained as a one-dimensional central extension of 4-dimensional metric Lie algebras endowed with a invertible parallel skew-symmetric tensor. On the other hand, we also classify $5$-dimensional metric Lie algebras with center of dimension greater than one admitting strict CKY tensors. In addition, we determine all possible CKY tensors on these metric Lie algebras. In particular, we exhibit the first…
A survey on invariant conformal Killing forms on Lie groups
本文综述了李群上左不变共形Killing p-形式的最新研究进展，并指出了未来研究中的有趣开放问题。
Invariant metrics with nonnegative curvature on compact Lie groups
我们对SO（3）和U（2）上具有非负曲率的左不变度量进行了分类.
In this paper we determine the moduli space, up to isometric automorphism, of left-invariant metrics on a $6$-dimensional Lie group $H$, such that its Lie algebra $\mathfrak{h}$ admits a complex structure and has first Betti number equal to four. We also investigate which of these metrics are Hermitian and classify the corresponding complex structures.
The moduli spaces of left-invariant statistical structures on Lie groups
本文研究了李群上左不变统计结构的模空间，针对三个特定李群，分析了其左不变黎曼度量的模空间，并分类了左不变共轭对称统计结构和双平坦结构。
We use the technique of invariant frames to study a left invariant spray structure on a Lie group, and calculate its S-curvature and Riemann curvature, which generalizes the corresponding formulae in homogeneous Finsler geometry. Using the canonical bi-invariant Berwald spray structure as the reference, any left invariant spray structure can be associated with a spray vector field on the Lie algebra. We find the correspondence between geodesics for a left invariant spray structure and the inverse integral curves of its spray vector field. As an application for this correspondence, we provide a…
The Ring of Invariants for Smooth Completions of Kac-Moody Lie Algebras
证明了Kac-Moody李代数的标准光滑完备的不变量环是由中心系数和Killing型两个元素生成的.
A strong KT (SKT) manifold consists of a Hermitian structure whose torsion three-form is closed. We classify the invariant SKT structures on four-dimensional solvable Lie groups. The classification includes solutions on groups that do not admit compact four-dimensional quotients. It also shows that there are solvable groups in dimension four that admit invariant complex structures but have no invariant SKT structure.
In the present report, by using the Stokes-Helmholtz decomposition theorem the 3-dimensional Navier-Stokes equation (NSE) is uncoupled and transformed into a scalar equation for the velocity potential when the flow field is toroidal. The dynamics of the velocity potential is independent of the vector potential. The reduction and invariant solutions to the equation are analyzed by the Lie symmetry method subsequently. The Lie subalgebras for the equation are discussed and the corresponding invariant solutions are also presented.
We determine the index of symmetry of 3-dimensional unimodular Lie groups with a left-invariant metric. In particular, we prove that every 3-dimensional unimodular Lie group admits a left-invariant metric with positive index of symmetry. We also study the geometry of the quotients by the so-called foliation of symmetry, and we explain in what cases the group fibers over a 2-dimensional space of constant curvature.
Techniques for classifying nonnegatively curved left-invariant metrics on compact Lie groups
我们提供了研究紧李群上的非负弯曲左不变度量的技巧。对于以双不变度量为起点，以非负曲率度量为终点的左不变度量的“直”路，我们推导出曲率的初导数的非负性质.我们应用这个结果得到SO（4）上非负曲左不变度量的部分分类。
For each connected and simply connected three-dimensional non-unimodular Lie group, we classify the left invariant Lorentzian metrics up to automorphism, and study the extent to which curvature can be altered by a change of metric. Thereby we obtain the Ricci operator, the scalar curvature, and the sectional curvatures as functions of left invariant Lorentzian metrics on the three-dimensional non-unimodular Lie groups. Our study is a continuation and extension of the previous studies done in \cite{HL2009_MN} for Riemannian metrics on three-dimensional Lie groups and in \cite{BC} for Lorentzian…
Hyperkähler torsion structures invariant by nilpotent Lie groups
本文研究了幂零李群上的HKT结构和相应的幂零流形上的HKT结构。我们在$\R^8$上证明了三个弱HKT结构，它们关于Heisenberg型李群的扩张是齐次的。相应的超复结构是一种特殊的类型，称为阿贝尔。我们证明了在任何2步幂零李群上，不变HKT结构都来自于阿贝尔超复结构。此外，我们使用交换超复结构和${\frak sp}(n)$的子空间之间的对应关系，产生连续族的紧和非紧的流形携带非等距HKT结构。最后，得到了2步幂零李群上不变HKT结构的几何性质。
On the ground of origins of the theory of Lie groups and Lie algebras, their (co)adjoint representations, and the Pontryagin maximum principle for the time-optimal problem are given an independent foundation for methods of geodesic vector field to search for normal geodesics of left-invariant (sub-)Finsler metrics on Lie groups and to look for the corresponding locally optimal controls in (sub-)Riemannian case, as well as some their applications.
It is known that a connected and simply-connected Lie group admits only one left-invariant Riemannian metric up to scaling and isometry if and only if it is isomorphic to the Euclidean space, the Lie group of the real hyperbolic space, or the direct product of the three dimensional Heisenberg group and the Euclidean space of dimension $n-3$. In this paper, we give a classification of left-invariant pseudo-Riemannian metrics of an arbitrary signature for the third Lie groups with $n \geq 4$ up to scaling and automorphisms. This completes the classifications of left-invariant pseudo-Riemannian m…
Left invariant nearly pseudo-Kähler structures and the tangent lie group
该微分几何研究证明，若李群上带有左不变近伪Kähler结构，则其带有诱导李群结构的切丛也存在左不变近伪Kähler结构，拓展了近伪Kähler结构的构造路径。
We show the correspondence between left invariant flat projective structures on Lie groups and certain prehomogeneous vector spaces. Moreover by using the classification theory of prehomogeneous vector spaces, we classify complex Lie groups admitting irreducible left invariant flat complex projective structures. As a result, direct sums of special linear Lie algebras sl(2) \oplus sl(m_1) \oplus \cdots \oplus sl(m_k) admit left invariant flat complex projective structures if the equality 4 + m_1^2 + \cdots + m_k^2 -k - 4 m_1 m_2 \cdots m_k = 0 holds. These contain sl(2), sl(2) \oplus sl(3)$, sl…
This is a survey work on Lie algebras with ad-invariant metrics. We summarize main features, notions and constructions, in the aim of bringing into consideration the main research on the topic. We also give some list of examples in low dimensions.
Left-invariant Kähler and semi-para-Kähler structures on some six-dimensional unsolvable Lie groups
针对李代数为半直积的六维不可解李群，本文依据四类李代数的分类，确定了左不变 Kähler、半 para-Kähler 及半 Kähler 结构的存在情形，并给出六维辛李代数可解性的例外结论。
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