Huibin Chen, Zhiqi Chen, Fuhai Zhu
In this paper, we focus on homogeneous spaces which are constructed from two strongly isotropy irreducible spaces, and prove that any geodesic orbit metric on these spaces is naturally reductive.
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The paper is devoted to the study of geodesic orbit Riemannian spaces that could be characterize by the property that any geodesic is an orbit of a 1-parameter group of isometries. The main result is the classification of compact simply connected geodesic orbit Riemannian spaces $G/H$ with two irreducible submodules in the isotropy representation.
We classify homogeneous pseudo-Riemannian manifolds of index 4 which admit an invariant almost hyper-Hermitian structure and an H-irreducible isotropy group. The main result is that all these spaces are flat except in dimension 12.
We show that pseudo-Riemannian almost quaternionic homogeneous spaces with index 4 and an H-irreducible isotropy group are locally isometric to a pseudo-Riemannian quaternionic Kähler symmetric space if the dimension is at least 16. In dimension 12 we give a non-symmetric example.
We show that for any weakly reflective submanifold of a compact isotropy irreducible Riemannian homogeneous space its inverse image under the parallel transport map is an infinite dimensional weakly reflective PF submanifold of a Hilbert space. This is an extension of the author's previous result in the case of compact irreducible Riemannian symmetric spaces. We also give a characterization of so obtained weakly reflective PF submanifolds.
A geodesic orbit manifold (GO manifold) is a Riemannian manifold (M,g) with the property that any geodesic in M is an orbit of a one-parameter subgroup of a group G of isometries of (M,g). The metric g is then called a G-GO metric in M. For an arbitrary compact homogeneous manifold M=G/H, we simplify the general problem of determining the G-GO metrics in M. In particular, if the isotropy representation of H induces equivalent irreducible submodules in the tangent space of M, we obtain algebraic conditions, under which, any G-GO metric in M admits a reduced form. As an application we determine …
Compact geodesic orbit spaces with a simple isotropy group
本文分类了以单 Lie 群为迷向群的紧致单连通黎曼齐性空间上的所有 $G$-自然约化度量和 $G$-测地轨道度量。
Riemannian geodesic orbit spaces (G/H,g) are natural generalizations of symmetric spaces, defined by the property that their geodesics are orbits of one-parameter subgroups of G. We study the geodesic orbit spaces of the form (G/S,g), where G is a compact, connected, semisimple Lie group and S is abelian. We give a simple geometric characterization of those spaces, namely that they are naturally reductive. In turn, this yields the classification of the invariant geodesic orbit (and also the naturally reductive) metrics on any space of the form G/S. Our approach involves simplifying the intrica…
Strongly isotropy irreducible spaces whose Laplacians have all their eigenspaces irreducible
本文针对强迷向不可约空间G/H，在算出拉普拉斯算子最小正特征值后，得到其第二个非零特征值，进而证明拉普拉斯算子所有特征空间均不可约的等价条件。
Invariant connections and $\nabla$-Einstein structures on isotropy irreducible spaces
本文研究非对称强迷向不可约齐性黎曼流形上的不变仿射、度量联络及带斜挠率的不变度量联络，计算其维数，分类紧李群${\ m U}_{n}$的双不变度量联络，并完成此类空间上带斜挠率的$\ abla$-Einstein不变结构的分类。
First Laplace eigenvalue of strongly isotropy irreducible spaces
该研究针对紧强迷向不可约空间的Laplace-Beltrami算子最小正特征值，给出单连通情形显式表达式，并证明其与Einstein常数满足$E<λ_1\leq 16E$的不等式关系。
We consider the Ricci flow equation for invariant metrics on compact and connected homogeneous spaces whose isotropy representation decomposes into two irreducible inequivalent summands. By studying the corresponding dynamical system, we completely describe the behaviour of the homogeneous Ricci flow on this kind of spaces. Moreover, we investigate the existence of ancient solutions and relate this to the existence and non-existence of invariant Einstein metrics.
In this paper, we classify compact simply connected cohomogeneity one manifolds up to equivariant diffeomorphism whose isotropy representation by the connected component of the principal isotropy subgroup has three or less irreducible summands. The manifold is either a bundle over a homogeneous space or an irreducible symmetric space. As a corollary such manifolds admit an invariant metric with non-negative sectional curvature.
Let $M$ be a most singular orbit of the isotropy representation of a simple symmetric space. Let $(ν_i, Φ_i)$ be an irreducible factor of the normal holonomy representation $(ν_pM, Φ(p))$. We prove that there exists a basis of a section $Σ_i\subset ν_i$ of $Φ_i$ such that the corresponding shape operators have rational eigenvalues (this is not in general true for other isotropy orbits). Conversely, this property, if referred to some non-transitive irreducible normal holonomy factor, characterizes the isotropy orbits. We also prove that the definition of a submanifold with constant principal cu…
We classify the $5$-dimensional homogeneous geometries in the sense of Thurston. The present paper (part 2 of 3) classifies those in which the linear isotropy representation is either irreducible or trivial. The $5$-dimensional geometries with irreducible isotropy are the irreducible Riemannian symmetric spaces, while those with trivial isotropy are simply-connected solvable Lie groups of the form $\mathbb{R}^3 \rtimes \mathbb{R}^2$ or $N \rtimes \mathbb{R}$ where $N$ is nilpotent.
We study the existence of invariant Einstein metrics on real flag manifolds associated to simple and non-compact split real forms of complex classical Lie algebras whose isotropy representation decomposes into two or three irreducible sub-representations. In this situation, one can have equivalent sub-modules, leading to the existence of non-diagonal homogeneous Riemannian metrics. In particular, we prove the existence of non-diagonal Einstein metrics on real flag manifolds.
We study geodesics in generalized Wallach spaces which are expressed as orbits of products of three exponential terms. These are homogeneous spaces $M=G/K$ whose isotropy representation decomposes into a direct sum of three submodules $\frak{m}=\frak{m}_1\oplus\frak{m}_2\oplus\frak{m}_3$, satisfying the relations $[\frak{m}_i,\frak{m}_i]\subset \frak{k}$. Assuming that the submodules $\frak{m}_i$ are pairwise non isomorphic, we study geodesics on such spaces of the form $γ(t)=\exp (tX)\exp (tY)\exp (tZ)\cdot o$, where $X\in\fr{m}_1, Y\in\fr{m}_2, Z\in\fr{m}_3$ ($o=eK$), with respect to a $G$-i…
Let M be an irreducible Riemannian symmetric space. The index i(M) of M is the minimal codimension of a totally geodesic submanifold of M. In previous work the authors proved that i(M) is bounded from below by the rank rk(M) of M. In this paper we classify all irreducible Riemannian symmetric spaces M for which the equality holds, that is, rk(M) = i(M). In this context we also obtain an explicit classification of all non-semisimple maximal totally geodesic submanifolds in irreducible Riemannian symmetric spaces of noncompact type and show that they are closely related to irreducible symmetric …
Homogeneous spaces with two equivalent isotropy summands develop positive Ricci curvature under Ricci flow
该研究针对具两个等价迷向子表示的单连通齐性空间，证明其G-不变度量经归一化里奇流后演化至正里奇曲率度量，且该正曲率度量族在流作用下前向不变。
We describe the invariant metrics on real flag manifolds and classify those with the following property: every geodesic is the orbit of a one-parameter subgroup. Such a metric is called g.o. (geodesic orbit). In contrast to the complex case, on real flag manifolds the isotropy representation can have equivalent submodules, which makes invariant metrics depend on more parameters and allows us to find more cases in which non-trivial g.o. metrics exist.
In this work it is shown that a necessary condition for the completeness of the geodesics of left invariant pseudo-Riemannian metrics on Lie groups is also sufficient in the case of 3-dimensional unimodular Lie groups, and not sufficient for 3-dimensional non unimodular Lie groups. As a consequence it is possible to identify, amongst the compact locally homogeneous Lorentzian 3-manifolds with non compact (local) isotropy group, those that are geodesically complete.
We classify maximal totally geodesic submanifolds in exceptional symmetric spaces up to isometry. Moreover, we introduce an invariant for certain totally geodesic embeddings of semisimple symmetric spaces, which we call the Dynkin index. We prove a result analogous to the index conjecture: for every irreducible symmetric space of non-compact type, there exists a totally geodesic submanifold which is of minimal codimension and whose non-flat irreducible factors have Dynkin index equal to one.
On the Prescribed Ricci Curvature of Noncompact Homogeneous Spaces with Two Isotropy Summands
本文对具有两个不可约求和项的单连通非紧齐性空间进行分类，并求解规定 Ricci 曲率问题。
On the Structure of Sub-Riemannian Geodesic Orbit Manifolds
该研究探讨齐次次黎曼流形的结构，证明其满足Goh条件且无严格异常长度极小化器，并分类了特定紧致连通单连通次黎曼测地线轨道流形，推广了相关黎曼分类成果。
We consider the question: can the isotropy representation of an irreducible pseudo-Riemannian symmetric space be realized as a conformal holonomy group? Using recent results of Cap, Gover and Hammerl, we study the representations of SO(2,1), PSU(2,1) and PSp(2,1) as isotropy groups of irreducible symmetric spaces of signature (3,2), (4,4) and (6,8), respectively, describing the geometry induced by a conformal holonomy reduction to the corresponding subgroups. In the case of SO(2,1) we show that conformal manifolds with such a holonomy reduction are always locally conformally flat and hence thi…
The paper is devoted to the study of geodesic orbit Riemannian spaces that could be characterize by the property that any geodesic is an orbit of a 1-parameter group of isometries. In particular, we discuss some important totally geodesic submanifolds that inherit the property to be geodesic orbit. For a given geodesic orbit Riemannian space, we describe the structure of the nilradical and the radical of the Lie algebra of the isometry group. In the final part, we discuss some new tools to study geodesic orbit Riemannian spaces, related to compact Lie group representations with non-trivial pri…
We construct a geodesic net in the plane with four unbalanced (boundary) vertices that has 16 balanced vertices and does not contain proper geodesic subnets. This is the first example of an irreducible geodesic net in the Euclidean plane with 4 boundary vertices that is not a tree.
A smooth curve on a homogeneous manifold $G/H$ is called a Riemannian equigeo-desic if it is a homogeneous geodesic for any $G$-invariant Riemannian metric. The homogeneous manifold $G/H$ is called Riemannian equigeodesic, if for any $x\in G/H$ and any nonzero $y\in T_x(G/H)$, there exists a Riemannian equigeodesic $c(t)$ with $c(0)=x$ and $\dot{c}(0)=y$. These two notions can be naturally transferred to the Finsler setting, which provides the definitions for Finsler equigeodesic and Finsler equigeodesic space. We prove two classification theorems for Riemannian equigeodesic spaces and Finsler…
We consider a homogeneous fibration $G/L \to G/K$, with symmetric fiber and base, where $G$ is a compact connected semisimple Lie group and $L$ has maximal rank in $G$. We suppose the base space $G/K$ is isotropy irreducible and the fiber $K/L$ is simply connected. We investigate the existence of $G$-invariant Einstein metrics on $G/L$ such that the natural projection onto $G/K$ is a Riemannian submersion with totally geodesic fibers. These spaces are divided in two types: the fiber $K/L$ is isotropy irreducible or is the product of two irreducible symmetric spaces. We classify all the $G$-inv…
Counting geodesics on compact symmetric spaces
本文通过焦轨道分解紧致对称空间指数映射的逆像，利用图、重数、Weyl群和格编码测地线的维数与连通分支，推广了紧致李群上的计数结果并给出已知结论的新证明。
Lorentzian homogeneous structures with indecomposable holonomy
本文研究洛伦兹齐性空间中迷向群的代数条件对几何与曲率的影响，证明具有不可分解、非不可约和乐群的Ambrose--Singer联络的洛伦兹局部齐性空间局部等距于平面波，推广了已有结果并给出关于平行挠率联络与2-对称联络的新结论。
We use the Cartan representations of $SO(3)$ and $SU(3)$, and an irreducible 14-dimensional representation of $Sp(3)$ to construct certain totally geodesic submanifolds in "skew" position in the complex quadrics, the complex 2-Grassmannians and the quaternionic 2-Grassmannians.
Totally geodesic submanifolds in products of rank one symmetric spaces
本文分类了秩一对称空间任意乘积中的全测地子流形，并在 Hermitian 对称空间中给出了无穷多个具有非平凡常 Kähler 角的不可约全测地子流形实例。
The present article is the final part of a series on the classification of the totally geodesic submanifolds of the irreducible Riemannian symmetric spaces of rank 2. After this problem has been solved for the 2-Grassmannians in my previous papers cited in the present paper as [K1] and [K2], and for the space SU(3)/SO(3) in [K3], Section 6, we now solve the classification for the remaining irreducible Riemannian symmetric spaces of rank 2 and compact type: SU(6)/Sp(3), SO(10)/U(5), E6/(U(1)*Spin(10)), E6/F4, G2/SO(4), SU(3), Sp(2) and G2. Similarly as for the spaces already investigated in the…
We continue the study of the distribution of closed geodesics on nilmanifolds constructed from a simply connected 2-step nilpotent Lie group with a left invariant metric and a lattice. We consider a Lie group with an associated 2-step nilpotent Lie algebra constructed from an irreducible representation of a compact semisimple Lie algebra on a real finite dimensional vector space. We determine sufficient conditions on the semisimple Lie algebra for the nilmanifold to have the density of closed geodesics property for a lattice arising from a Chevalley rational structure.
On Closed 6-Manifolds Admitting Riemannian Metrics with Positive Sectional Curvature and Non-Abelian Symmetry
该文研究具有正截面曲率且容许SU(2)或SO(3)等距作用的闭单连通6维黎曼流形，确定其欧拉示性数与已知例子一致，分类相关作用并给出例外层限制。
Homogeneous Einstein metrics on generalized flag manifolds with five isotropy summands
该研究针对具有五个迷向和项的紧单李群广义旗流形，结合旗流形纤维化与黎曼淹没技术构建齐性爱因斯坦方程，通过Gröbner基分类流形并求解，得到部分流形的全部不变爱因斯坦度量及部分流形的非Kähler-Einstein度量存在性结论。
We investigate homogeneous geodesics in a class of homogeneous spaces called $M$-spaces, which are defined as follows. Let $G/K$ be a generalized flag manifold with $K=C(S)=S\times K_1$, where $S$ is a torus in a compact simple Lie group $G$ and $K_1$ is the semisimple part of $K$. Then the {\it associated $M$-space} is the homogeneous space $G/K_1$. These spaces were introduced and studied by H.C. Wang in 1954. We prove that for various classes of $M$-spaces the only g.o. metric is the standard metric. For other classes of $M$-spaces we give either necessary, or necessary and sufficient condi…
We show that if $M$ is an Einstein hypersurface in an irreducible Riemannian symmetric space $\overline{M}$ of rank greater than $1$ (the classification in the rank-one case was previously known), then either $\overline{M}$ is of noncompact type and $M$ is a codimension one Einstein solvmanifold, or $\overline{M}=\mathrm{SU}(3)/\mathrm{SO}(3)$ (respectively, $\overline{M}=\mathrm{SL}(3)/\mathrm{SO}(3)$) and $M$ is foliated by totally geodesic spheres (respectively, hyperbolic planes) of $\overline{M}$, with the space of leaves parametrised by a special Legendrian surface in $S^5$ (respectively…
Let M be an irreducible Riemannian symmetric space. The index i(M) of M is the minimal codimension of a (non-trivial) totally geodesic submanifold of M. The purpose of this note is to determine the index i(M) for all irreducible Riemannian symmetric spaces M of type (II) and (IV).
We study geodesically complete and locally compact Hadamard spaces X whose Tits boundary is a connected irreducible spherical building. We show that X is symmetric iff complete geodesics in X do not branch and a Euclidean building otherwise. Furthermore, every boundary equivalence (cone topology homeomorphism preserving the Tits metric) between two such spaces is induced by a homothety. As an application, we can extend the Mostow and Prasad rigidity theorems to compact singular (orbi)spaces of nonpositive curvature which are homotopy equivalent to a quotient of a symmetric space or Euclidean b…
Let M be an irreducible Riemannian symmetric space. The index of M is the minimal codimension of a (non-trivial) totally geodesic submanifold of M. We prove that the index is bounded from below by the rank of the symmetric space. We also classify the irreducible Riemannian symmetric spaces whose index is less or equal than 3.
Metric operator and geodesic orbit property for a standard homogeneous Finsler metric
本文为紧致齐次Finsler空间引入度量算子，分类了具有两个不可约和项且存在g.o.非自然约化标准齐次(α1,α2)-度量的流形，并证明广义Wallach空间上标准齐次(α1,α2,α3)-度量g.o.当且仅当为正规齐次黎曼度量。
In this work we study the existence of homogeneous Einstein metrics on the total space of homogeneous fibrations such that the fibers are totally geodesic manifolds. We obtain the Ricci curvature of an invariant metric with totally geodesic fibers and some necessary conditions for such a metric to be Einstein in terms of Casimir operators. Some particular cases are studied, for instance, for normal base or fiber, symmetric fiber, Einstein base or fiber, for which the Einstein equations are manageable. We show the existence of new invariant Einstein metrics on homogeneous bisymmetric fibrations…
A Unified Construction of Exceptional Symmetric Spaces
该研究给出了所有单连通不可约例外紧对称空间的统一构造，通过确定其包含关系得到全测地嵌入的统一描述，发现其包含结构具意外对称性，揭示了例外对称空间间的深层几何关联。
Let $M$ be an irreducible Hermitian symmetric space of compact type, and let $ω$ be its Kähler form. For a triplet $(p_1,p_2,p_3)$ of points in $M$ we study conditions under which a geodesic triangle $\mathcal T(p_1,p_2,p_3)$ with vertices $p_1,p_2,p_3$ can be unambiguously defined. We consider the integral $A(p_1,p_2,p_3)=\int_Σω$, where $Σ$ is a surface filling the triangle $\mathcal T(p_1,p_2,p_3)$ and discuss the dependence of $A(p_1,p_2,p_3)$ on the surface $Σ$. Under mild conditions on the three points, we prove an explicit formula for $A(p_1,p_2,p_3)$ analogous to the known formula for …
A groupoidal description of elementary particles
本文通过扩展时空对称性的描述框架至更灵活的运动群束，扩展了威金的程序至弯曲时空。提出基本粒子为运动群束的不可约项目表示，通过自然的威金群束定义，展示了这些表示具有与庞加莱群相似的量子数，并通过扩展麦克凯理论分类了基本粒子。
For a compact homogeneous space $G/K$, we study the problem of existence of $G$-invariant Riemannian metrics such that each eigenspace of the Laplacian is a real irreducible representation of $G$. We prove that the normal metric of a compact irreducible symmetric space has this property only in rank one. Furthermore, we provide existence results for such metrics on certain isotropy reducible spaces.
Let $G/H$ be a compact homogeneous space, and let $\hat{g}_0$ and $\hat{g}_1$ be $G$-invariant Riemannian metrics on $G/H$. We consider the problem of finding a $G$-invariant Einstein metric $g$ on the manifold $G/H\times [0,1]$ subject to the constraint that $g$ restricted to $G/H\times \{0\}$ and $G/H\times \{1\}$ coincides with $\hat{g}_0$ and $\hat{g}_1$, respectively. By assuming that the isotropy representation of $G/H$ consists of pairwise inequivalent irreducible summands, we show that we can always find such an Einstein metric.
We study homogeneous curves in generalized flag manifolds $G/K$ with $G_2$-type $t$-roots, which are geodesics with respect to each $G$-invariant metric on $G/K$. These curves are called equigeodesics. The tangent space of such flag manifolds splits into six isotropy summands, which are in one-to-one correspondence with $t$-roots. Also, these spaces are a generalization of the exceptional full flag manifold $G_2/T$. We give a characterization for structural equigeodesics for flag manifolds with $G_2$-type $t$-roots, and we give for each such flag manifold, a list of subspaces in which the vect…
On cohomogeneity one hyperpolar actions related to $G_{2}$
本文针对紧型不可约黎曼对称空间上与例外紧李群$G_{2}$相关的余维数1例外作用，研究其轨道作为黎曼子流形的性质，考察主轨道主曲率并分类极小等类型主轨道。
We prove that a function on an irreducible compact symmetric space M, which is not a sphere, is determined by its integrals over the shortest closed geodesics in M. We also prove a support theorem for the Funk transform on rank one symmetric spaces which are not spheres.
Sasaki-Einstein orbits in compact Hermitian symmetric spaces
本文研究紧型不可约Hermitian对称空间上迷向子群轨道的CR子流形性质，证明叶为轨道全测地子流形，探讨其成为环境空间全测地子流形的条件，分类具接触或Sasakian结构的轨道并确定Sasaki-Einstein轨道。
We classify non-polar irreducible representations of connected compact Lie groups whose orbit space is isometric to that of a representation of a finite extension of $Sp(1)^k$ for some $k>0$. It follows that they are obtained from isotropy representations of certain quaternion-Kähler symmetric spaces by restricting to the "non-$Sp(1)$-factor".
The classification of Riemannian manifolds by the holonomy group of their Levi-Civita connection picks out many interesting classes of structures, several of which are solutions to the Einstein equations. The classification has two parts. The first consists of isolated examples: the Riemannian symmetric spaces. The second consists of geometries that can occur in continuous families: these include the Calabi-Yau structures and Joyce manifolds of string theory. One may ask how one can weaken the definitions and still obtain similar classifications. We present two closely related suggestions. The…
Back in 1985, Wang and Ziller obtained a complete classification of all homogeneous spaces of compact simple Lie groups on which the standard or Killing metric is Einstein. The list consists, beyond isotropy irreducible spaces, of 12 infinite families (two of them are actually conceptual constructions) and 22 isolated examples. We study in this paper the nature of each of these Einstein metrics as a critical point of the scalar curvature functional.
Cohomogeneity one Ricci Solitons from Hopf Fibrations
本文针对主轨道迷向表示含两个不可约子和的情形，证明非平凡丛上存在非相似完备稳态与膨胀余齐次性一 Ricci 孤立子，并用于重构 Böhm 的正标量曲率 Einstein 度量。
In this paper, we generalize Medos-Wang's arguments and results on the mean curvature flow deformations of symplectomorphisms of $\CP^n$ in \cite{MeWa} to complex Grassmann manifold $G(n, n+m;\C)$ and compact totally geodesic Kähler-Einstein submanifolds of $G(n, 2n;\C)$ such as irreducible Hermitian symmetric spaces $SO(2n)/U(n)$ and $Sp(n)/U(n)$ (in the terminology of \cite[p. 518]{He}). We also give an abstract result and discuss the case of complex tori.
The classification of homogeneous Einstein metrics on flag manifolds with $b_2(M)=1$
本文针对第二贝蒂数为1的广义旗流形中两个未分类的例外E8型旗流形，用黎曼淹没方法构建爱因斯坦方程并完成齐性爱因斯坦度量分类，最终证明这类流形的非凯勒不变爱因斯坦度量数量有限。
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…
Let $D_F = \{(z_0, z) \in {\C}^{n} | |z_0|^2 < b, \|z\|^2 < F(|z_0|^2) \}$ be a strongly pseudoconvex Hartogs domain endowed with the \K metric $g_F$ associated to the \K form $ω_F = -\frac{i}{2} \partial \bar{\partial} \log (F(|z_0|^2) - \|z\|^2)$. This paper contains several results on the Riemannian geometry of these domains. In the first one we prove that if $D_F$ admits a non special geodesic (see definition below) through the origin whose trace is a straight line then $D_F$ is holomorphically isometric to an open subset of the complex hyperbolic space. In the second theorem we prove that…
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