Richard Cushman
This note shows that the module of smooth vector fields on ${\mathbb{R}}^n$, which are invariant under the linear action of a compact Lie group $G$ is finitely generated by polynomial vector fields on ${\mathbb{R}}^n$ which are invariant under the action of $G$.
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We introduce the notion of a subregular subalgebra, which we believe is useful for classification of subalgebras of Lie algebras. We use it to construct a non-regular invariant generalized complex structure on a Lie group. As an illustration of the study of invariant generalized complex structures, we compute them all for the real forms of G2.
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.
The aim of this paper is to determine left-invariant strictly almost Kähler structures on 4-dimensional Lie groups $(g, J, Ω)$ such that the Ricci tensor is $J$-invariant.
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.
On simple compact Lie skew braces
本文从全局和微分视角研究李斜 brace 的简单性，证明紧致连通简单李斜 brace 的结构特性，并构造非紧致例子展示其非刚性。
$k$-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…
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.
The purpose of this paper is to show that any extension of a minimal Lie foliation on a compact manifold is a transversaly Riemannian g\h- foliation with trivial normal bundle. This result permits to classify the extensions of a minimal Lie foliation on a compact manifold from the Lie subgroups of its Lie group.
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。
The main result of this paper is the classification of the real irreducible representations of compact Lie groups with vanishing homogeneity rank.
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 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.
We give a classification, up to local isomorphisms, of semi-simple Lie groups without compact factors that can act faithfully and conformally on a compact Lorentz manifold of dimension greater than or equal to $3$.
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.
In this paper we classify the reducible representations of compact simple Lie groups all of whose orbits are tautly embedded in Euclidean space with respect to Z_2 coefficients.
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.
In this paper we study (smooth and holomorphic) foliations which are invariant under transverse actions of Lie groups.
We give a new method for manufacturing complete minimal submanifolds of compact Lie groups and their homogeneous quotient spaces. For this we make use of harmonic morphisms and basic representation theory of Lie groups. We then apply our method to construct many examples of compact minimal submanifolds of the special unitary groups.
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。
We show that, in compact semisimple Lie groups and Lie algebras, any neighbourhood of the identity gets mapped, under the commutator map, to a neighbourhood of the identity.
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.
Based on the representation theory and the study on the involutions of compact simple Lie groups, we show that $F_4$ admits non-naturally reductive Einstein metrics.
Let $G$ be a connected, simply-connected, compact simple Lie group. In this paper, we show that the isometry group of $G$ with a left-invariant pseudo-Riemannan metric is compact. Furthermore, the identity component of the isometry group is compact if $G$ is not simply-connected.
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.
Semi-Riemannian metrics on compact simple Lie Groups
该论文是一篇针对紧李群上左不变半黎曼度量相关研究的综述性文章。
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 show that a left invariant metric on a compact Lie group $G$ which is obtained by stretching a biinvariant metric in the direction of a subalgebra $\h$ of $\g$ always has some negative sectional curvature, unless the semi-simple part of $\h$ is an ideal of $\g$.
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。
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.
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。
Complete Minimal Submanifolds of the Non-Compact Duals to the Classical Compact Lie Groups
该研究针对经典紧李群的非紧对偶空间，构造了新的余维数2的完备极小子流形族，丰富了极小子流形的相关研究成果。
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…
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.
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.
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.
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…
We study HKT structures on nilpotent Lie groups and on associated nilmanifolds. We exhibit three weak HKT structures on $\R^8$ which are homogeneous with respect to extensions of Heisenberg type Lie groups. The corresponding hypercomplex structures are of a special kind, called abelian. We prove that on any 2-step nilpotent Lie group all invariant HKT structures arise from abelian hypercomplex structures. Furthermore, we use a correspondence between abelian hypercomplex structures and subspaces of ${\frak sp}(n)$ to produce continuous families of compact and noncompact of manifolds carrying no…
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。
Several new invariants for Lie algebroids have been discovered recently. We give an overview of these invariants and establish several relationships between them.
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