The distribution of symmetry of a naturally reductive nilpotent Lie group

Naturally reductive pseudo-Riemannian spaces

A family of naturally reductive pseudo-Riemannian spaces is constructed out of the representations of Lie algebras with ad-invariant metrics. We exhibit peculiar examples, study their geometry and characterize the corresponding naturally reductive homogeneous structure.

Non-naturally reductive Einstein metrics on the compact simple Lie group $F_4$

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.

Four-dimensional naturally reductive pseudo-Riemannian spaces

The classification of 4-dimensional naturally reductive pseudo-Riemannian spaces is given. This classification comprises symmetric spaces, the product of 3-dimensional naturally reductive spaces with the real line and new families of indecomposable manifolds which are studied at the end of the article. The oscillator group is also analyzed from the point of view of this classification.

自然约化齐性 $(\alpha,\beta)$-度量空间

Naturally reductive homogeneous $(α,β)$-metric spaces

本文研究自然约化齐性 $(\alpha,\beta)$-度量空间，给出其自然约化的充要条件，证明两种定义等价，并计算旗曲率。

Non-naturally reductive Einstein metrics on $\mathrm{SO}(n)$

In this article, we prove that every compact simple Lie group $SO(n)$ for $n\geq 10$ admits at least $2\left([\frac{n-1}{3}]-2\right)$ non-naturally reductive left-invariant Einstein metrics.

Examples of naturally reductive pseudo-Riemannian Lie groups

We provide examples of naturally reductive pseudo-Riemannian spaces, in particular an example of a naturally reductive pseudo-Riemannian 2-step nilpotent Lie group $(N, < \,,\,>_N)$, such that $< \,,\,>_N$ is invariant under a left action and for which the center is degenerate. The metric does not correspond to a bi-invariant one.

Geodesic orbit and naturally reductive nilmanifolds associated with graphs

We study Riemannian nilmanifolds associated with graphs. We prove that such a nilmanifold is geodesic orbit if and only if it is naturally reductive if and only if its defining graph is the disjoint union of complete graphs and the left-invariant metric is generated by a certain naturally defined inner product.

Einstein metrics on compact Lie groups which are not naturally reductive

The study of left-invariant Einstein metrics on compact Lie groups which are naturally reductive was initiated by J. E. D'Atri and W. Ziller in 1979. In 1996 the second author obtained non-naturally reductive Einstein metrics on the Lie group SU(n) for $n \ge 6$, by using a method of Riemannian submersions. In the present work we prove existence of non-naturally reductive Einstein metrics on the compact simple Lie groups SO(n) ($n \geq 11$), $Sp(n)$ ($n \geq 3$), $E_6$, $E_7$, and $E_8$.

洛伦兹自然约化幂零流形的对称性分布

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。

Homogeneous geodesics of non-unimodular Lorentzian Lie groups and naturally reductive Lorentzian spaces in dimension three

We determine, for all three-dimensional non-unimodular Lie groups equipped with a Lorentzian metric, the set of homogeneous geodesics through a point. Together with the results of [C] and [CM2], this leads to the full classification of three-dimensional Lorentzian g.o. spaces and naturally reductive spaces.

Non-naturally reductive Einstein metrics on normal homogeneous Einstein manifolds

It is an important problem in differential geometry to find non-naturally reductive homogeneous Einstein metrics on homogeneous manifolds. In this paper, we consider this problem for some coset spaces of compact simple Lie groups. A new method to construct invariant non-naturally reductive Einstein metrics on normal homogeneous Einstein manifolds is presented. In particular, we show that on the standard homogeneous Einstein manifolds, except for some special cases, there exist plenty of such metrics. A further interesting result of this paper is that on some compact semisimple Lie groups, ther…

紧李群上自然约化度量的预定里奇曲率问题

The prescribed Ricci curvature problem for naturally reductive metrics on compact Lie groups

本文研究紧李群上自然约化度量类中的预定里奇曲率问题，给出了方程可解的必要与充分条件，并通过一系列例子加以说明。

Jacobi relations on naturally reductive homogeneous spaces

Naturally reductive spaces, in general, can be seen as an adequate generalization of Riemannian symmetric spaces. Nevertheless, there are some that are closer to symmetric spaces than others. On the one hand, there is the series of Hopf fibrations over complex space forms, including the Heisenberg groups with their metrics of type H. On the other hand, there exist certain naturally reductive spaces in dimensions six and seven whose torsion forms have a distinguished algebraic property. All these spaces generalize geometric or algebraic properties of $3$--dimensional naturally reductive spaces …

Tangent Lie groups are Riemannian naturally reductive spaces

Given a compact Lie group $G$ with Lie algebra $\mathfrak{g}$, we consider its tangent Lie group $TG\cong G\ltimes_{\mathrm{Ad}} \mathfrak{g}$. In this short note, we prove that $TG$ admits a left-invariant naturally reductive Riemannian metric $g$ and a metric connection with skew torsion $\nabla$ such that $(TG,g,\nabla)$ is naturally reductive. An alternative spinorial description of the same connection on the direct product $G\times \mathfrak{g}$ generalizes in a rather subtle way to $TS^7$, which is in many senses almost a tangent Lie group.

Non-naturally reductive Einstein metrics on exceptional Lie groups

Given an exceptional compact simple Lie group $G$ we describe new left-invariant Einstein metrics which are not naturally reductive. In particular, we consider fibrations of $G$ over flag manifolds with a certain kind of isotropy representation and we construct the Einstein equation with respect to the induced left-invariant metrics. Then we apply a technique based on Gröbner bases and classify the real solutions of the associated algebraic systems. For the Lie group ${\rm G}_2$ we obtain the first known example of a left-invariant Einstein metric, which is not naturally reductive. Moreover, f…

On the smallest Laplace eigenvalue for naturally reductive metrics on compact simple Lie groups

Eldredge, Gordina and Saloff-Coste recently conjectured that, for a given compact connected Lie group $G$, there is a positive real number $C$ such that $λ_1(G,g)\operatorname{diam}(G,g)^2\leq C$ for all left-invariant metrics $g$ on $G$. In this short note, we establish the conjecture for the small subclass of naturally reductive left-invariant metrics on a compact simple Lie group.

非紧单李群上自然约化度量的预定 Ricci 曲率问题

The prescribed Ricci curvature problem for naturally reductive metrics on non-compact simple Lie groups

本文研究了非紧单李群上左不变自然约化 Riemannian 度量的预定 Ricci 曲率问题，给出了底层方程可解性的若干条件并讨论了相关例子。

A note on the uniqueness of the canonical connection of a naturally reductive space

We extend the result in J. Reine Angew. Math. 664, 29-53, to the non-compact case. Namely, we prove that the canonical connection on a simply connected and irreducible naturally reductive space is unique, provided the space is not a sphere, a compact Lie group with a bi-invariant metric or its symmetric dual. In particular, the canonical connection is unique for the hyperbolic space when the dimension is different from three. We also prove that the canonical connection on the sphere is unique for the symmetric presentation. Finally, we compute the full isometry group (connected component) of a…

New Einstein metrics on the Lie group $SO(n)$ which are not naturally reductive

We obtain new invariant Einstein metrics on the compact Lie groups $SO(n)$ ($n \geq 7$) which are not naturally reductive. This is achieved by imposing certain symmetry assumptions in the set of all left-invariant metrics on $SO(n)$ and by computing the Ricci tensor for such metrics. The Einstein metrics are obtained as solutions of systems polynomial equations, which we manipulate by symbolic computations using Gröbner bases.

New Non-naturally reductive Einstein metrics on Exceptional simple Lie groups

In this article, we achieved several non-naturally reductive Einstein metrics on exceptional simple Lie groups, which are formed by the decomposition arising from general Wallach spaces. By using the decomposition corresponding to the two involutive automorphisms, we calculated the non-zero coefficients in the expression for the components of Ricci tensor with respect to the given metrics. The Einstein metrics are obtained as solutions of systems polynomial equations, which we manipulate by symbolic computations using Gröbner bases.

循环黎曼幂零流形是自然约化的

Cyclic Riemannian nilmanifolds are naturally reductive

本文研究循环黎曼幂零流形，证明其循环性等价于李代数至多两步幂零且相关斜对称族可交换，从而推出自然约化性，并给出无欧几里得因子时的完全等距群。

借助实旗流形得到的正交群上非自然约化爱因斯坦度量

Non naturally reductive Einstein metrics on the orthogonal group via real flag manifolds

本文通过利用实旗流形并对左不变度量施加对称性假设，在紧李群$\SO(n)$上构造出非自然约化的新不变爱因斯坦度量。

The index of symmetry of compact naturally reductive spaces

We introduce a geometric invariant that we call the index of symmetry, which measures how far is a Riemannian manifold from being a symmetric space. We compute, in a geometric way, the index of symmetry of compact naturally reductive spaces. In this case, the so-called leaf of symmetry turns out to be of the group type. We also study several examples where the leaf of symmetry is not of the group type. Interesting examples arise from the unit tangent bundle of the sphere of curvature 2, and two metrics in an Aloff-Wallach 7-manifold and the Wallach 24-manifold.

Finsler几何中基于测地线图的自然还原性

The natural reductivity in Finsler geometry in terms of geodesic graphs

本文提出自然还原Finsler流形的测地线图新定义，构造纯Finsler自然还原度量例子，并描述非自然还原度量的测地线图及单形式影响。

Naturally reductive pseudo-Riemannian Lie groups in low dimensions

This work concerns the non-flat metrics on the Heisenberg Lie group of dimension three $\Heis_3(\RR)$ and the bi-invariant metrics on the solvable Lie groups of dimension four. On $\Heis_3(\RR)$ we prove that the property of the metric being naturally reductive is equivalent to the property of the center being non-degenerate. These metrics are Lorentzian algebraic Ricci solitons. We start with the indecomposable Lie groups of dimension four admitting bi-invariant metrics and which act on $\Heis_3(\RR)$ by isometries and we finally study some geometrical features on these spaces.

On some classes of Z-graded Lie algebras

We study finite dimensional almost and quasi-effective prolongations of nilpotent Z-graded Lie algebras, especially focusing on those having a decomposable reductive structural subalgebra. Our assumptions generalize effectiveness and algebraicity and are appropriate to obtain Levi-Malčev and Levi-Chevalley decompositions and precisions on the heigth and other properties of the prolongations in a very natural way. In a last section we systematically present examples in which simple Lie algebras are obtained as prolongations, for reductive structural algebras of type A, B, C and D, of nilpotent …

Quaternionic Heisenberg groups as naturally reductive homogeneous spaces

In this note, we describe the geometry of the quaternionic Heisenberg groups from a Riemannian viewpoint. We show, in all dimensions, that they carry an almost $3$-contact metric structure which allows us to define the metric connection that equips these groups with the structure of a naturally reductive homogeneous space. It turns out that this connection, which we shall call the canonical connection because of its analogy to the $3$-Sasaki case, preserves the horizontal and vertical distributions and even the quaternionic contact structure of the quaternionic Heisenberg groups. We focus on t…

自然约化性质的不可听性

Inaudibility of naturally reductive property

本文通过构造9维2步幂零流形的等谱对（一个自然约化、一个非自然约化），结合Ambrose-Singer齐次结构刻画，证明无法利用拉普拉斯-贝尔特拉米算子谱判断闭黎曼流形是否自然约化。

The skew-torsion holonomy theorem and naturally reductive spaces

We prove a Simons-type holonomy theorem for totally skew 1-forms with values in a Lie algebra of linear isometries. The only transitive case, for this theorem, is the full orthogonal group. We only use geometric methods and we do not use any classification (not even that of transitive isometric actions on the sphere or the list of rank one symmetric spaces). This result was independently proved, by using an algebraic approach, by Paul-Andy Nagy. We apply this theorem to prove that the canonical connection of a compact naturally reductive space is unique, provided the space does not split off, …

通过广义旗流形构造 $\SU(N)$ 上的非自然约化 Einstein 度量

Non naturally reductive Einstein metrics on $\SU(N)$ via generalized flag manifolds

本文利用广义旗流形及对 $\frak k$ 中心正交基的对称性选择，在紧 Lie 群 $\SU(N)$ 上构造了新的非自然约化不变 Einstein 度量，并讨论其等距性。

Jacobi osculating rank and isotropic geodesics on naturally reductive 3-manifolds

We study the Jacobi osculating rank of geodesics on naturally reductive homogeneous manifolds and we apply this theory to the 3-dimensional case. Here, each non-symmetric, simply connected naturally reductive 3-manifold can be given as a principal bundle over a surface of constant curvature, such that the curvature of its horizontal distribution is also a constant. Then, we prove that the Jacobi osculating rank of every geodesic is two except for the Hopf fibers, where it is zero. Moreover, we determine all isotropic geodesics and the isotropic tangent conjugate locus.

Spectral isolation of naturally reductive metrics on simple Lie groups

We show that within the class of left-invariant naturally reductive metrics $\mathcal{M}_{\operatorname{Nat}}(G)$ on a compact simple Lie group $G$, every metric is spectrally isolated. We also observe that any collection of isospectral compact symmetric spaces is finite; this follows from a somewhat stronger statement involving only a finite part of the spectrum.

On invariant Riemannian metrics on Ledger-Obata spaces

We study invariant metrics on Ledger-Obata spaces $F^m/\mathrm{diag}(F)$. We give the classification and an explicit construction of all naturally reductive metrics, and also show that in the case $m=3$, any invariant metric is naturally reductive. We prove that a Ledger-Obata space is a geodesic orbit space if and only if the metric is naturally reductive. We then show that a Ledger-Obata space is reducible if and only if it is isometric to the product of Ledger-Obata spaces (and give an effective method of recognising reducible metrics), and that the full connected isometry group of an irred…

自然约化性质的不可闻性（非紧情形）

Non-compact inaudibility of Naturally Reductive property

本文利用一对11维非紧广义海森堡群，研究2步幂零李群与自然约化性质的关系，证明该性质在非紧情形下是不可闻的。

2-步幂零李群上的G$_2$-瞬子

G$_2$-instantons on $2$-step nilpotent Lie groups

本文在7维2-步幂零李群上研究由特征联络产生的一族度量联络的G$_2$-瞬子条件，根据换位子子群维数给出充分必要条件，并证明这些瞬子诱导自然约化结构，从而通过取格商得到紧致幂零流形上的G$_2$-瞬子。

On Lorentzian connections with parallel skew torsion

The paper is devoted to metric connections with parallel skew-symmetric torsion in Lorentzian signature. This is motivated by recent progress in the Riemannian signature and by possible applications to supergravity theories. We provide a complete information about holonomy algebras, torsion and curvature of the considered connections up to the corresponding objects from the Riemannian signature. Various examples are constructed. It is shown how to construct all simply connected Lorentzian naturally reductive homogeneous spaces of arbitrary dimension from Riemannian naturally reductive homogene…

On nonsingular two-step nilpotent Lie algebras

A 2-step nilpotent Lie algebra n is called nonsingular if ad(X): n --> [n,n] is onto for any X not in [n,n]. We explore nonsingular algebras in several directions, including the classification problem (isomorphism invariants), the existence of canonical inner products (nilsolitons) and their automorphism groups (maximality properties). Our main tools are the moment map for certain real reductive representations, and the Pfaffian form of a 2-step algebra, which is a positive homogeneous polynomial in the nonsingular case.

自然红uctive齐性空间上的拉普拉斯-贝叶斯谱

The Laplace-Beltrami spectrum on Naturally Reductive Homogeneous Spaces

本文研究了紧致自然红uctive齐性空间上拉普拉斯-贝叶斯算子的谱，通过广义Casimir算子的特征值和球形表示推导出公式，并应用于正常齐性度量的变体，首次分析了谱对度量变形参数的依赖性。

The classification of naturally reductive homogeneous spaces in dimensions $n \leq 6$

We present a new method for classifying naturally reductive homogeneous spaces -- i.\,e.~homogeneous Riemannian manifolds admitting a metric connection with skew torsion that has parallel torsion \emph{and} curvature. This method is based on a deeper understanding of the holonomy algebra of connections with parallel skew torsion on Riemannian manifolds and the interplay of such a connection with the geometric structure on the given Riemannian manifold. It allows to reproduce by easier arguments the known classifications in dimensions $3,4$, and $5$, and yields as a new result the classificatio…

Notes on "Einstein metrics on compact simple Lie groups attached to standard triples"

In the paper "Einstein metrics on compact simple Lie groups attached to standard triples", the authors introduced the definition of standard triples and proved that every compact simple Lie group $G$ attached to a standard triple $(G,K,H)$ admits a left-invariant Einstein metric which is not naturally reductive except the standard triple $(\Sp(4),2\Sp(2),4\Sp(1))$. For the triple $(\Sp(4),2\Sp(2),4\Sp(1))$, we find there exists an involution pair of $\sp(4)$ such that $4\sp(1)$ is the fixed point of the pair, and then give the decomposition of $\sp(4)$ as a direct sum of irreducible $\ad(4\sp(…

Invariant Einstein metrics on SU(N) and complex Stiefel manifolds

We study existence of invariant Einstein metrics on complex Stiefel manifolds $G/K = \SU(\ell+m+n)/\SU(n) $ and the special unitary groups $G = \SU(\ell+m+n)$. We decompose the Lie algebra $\frak g$ of $G$ and the tangent space $\frak p$ of $G/K$, by using the generalized flag manifolds $G/H = \SU(\ell+m+n)/\s(\U(\ell)\times\U(m)\times\U(n))$. We parametrize scalar products on the 2-dimensional center of the Lie algebra of $H$, and we consider $G$-invariant and left invariant metrics determined by $\Ad(\s(\U(\ell)\times\U(m)\times\U(n))$-invariant scalar products on $\frak g$ and $\frak p$ res…

Connections on naturally reductive spaces, their Dirac operator and homogeneous models in string theory

Given a reductive homogeneous space M=G/H endowed with a naturally reductive metric, we study the one-parameter family of connections joining the canonical and the Levi-Civita connection (t=0, 1/2). We show that the Dirac operator D^t corresponding to t=1/3 is the so-called ``cubic'' Dirac operator recently introduced by B. Kostant, and derive the formula for its square for any t, thus generalizing the classical Parthasarathy formula on symmetric spaces. Applications include the existence of a new G-invariant first order differential operator on spinors and an eigenvalue estimate for the first…

Detecting orbits along subvarieties via the moment map

Let G be a (real or complex) linear reductive algebraic group acting on an affine variety V. Let W be a subvariety. In this work we study how the G-orbits intersect W. We develop a criterion to determine when the intersection can be described as a finite union of orbits of a reductive subgroup. The conditions of the criterion are easily verified in practice and are used to construct continuous families of (non-isomorphic) nilpotent Lie groups which do not admit left-invariant Ricci soliton metrics. Other applications to the left-invariant geometry of Lie groups are also given. The note finishe…

Plancherel Formulae associated to Filtrations of Nilpotent Lie Groups

We study the conditions for a nilpotent Lie group to be foliated into subgroups that have square integrable (relative discrete series) unitary representations, that fit together to form a filtration by normal subgroups. Then we use that filtration to construct a class of "stepwise square integrable" representations on which Plancherel measure is concentrated. Further, we work out the character formulae for those stepwise square integrable representations, and we give an explicit Plancherel formula. Next, we use some structure theory to check that all these constructions and results apply to ni…

On the existence of nilsolitons on 2-step nilpotent Lie groups

A 2-step nilpotent Lie algebra n is said to be of type (p,q)if dim(n)=p+q and dim([n,n])=p. By considering a class of 2-step nilpotent Lie algebras naturally attached to graphs, we prove that there exist indecomposable, 2-step nilpotent Lie groups of type (p,q) which do not admit a nilsoliton metric for every pair (p,q) such that 20 < q and q-2 < p < 1/2q^2-5/2q+10. This improves a result due to Jablonski.

Geometric structures on nilpotent Lie groups: on their classification and a distinguished compatible metric

Let N be a nilpotent Lie group and let S be an invariant geometric structure on N (cf. symplectic, complex or hypercomplex). We define a left invariant Riemannian metric on N compatible with S to be "minimal", if it minimizes the norm of the invariant part of the Ricci tensor among all compatible metrics with the same scalar curvature. We prove that minimal metrics (if any) are unique up to isometry and scaling, they develop soliton solutions for the invariant Ricci flow and are characterized as the critical points of a natural variational problem. The uniqueness allows us to distinguish geome…

两步幂零李群上的复结构

Complex structures on two-step nilpotent Lie groups

本文刻画了容许左不变复结构的 $2$ 步幂零李代数，研究了其 Hermitian 几何，给出了度量为 pluriclosed 的充要条件，并将结果推广至 hypercomplex 情形。