arXiv math/06046082006-04math.DG

Hermitian structures on cotangent bundles of four dimensional solvable Lie groups

L. C. de Andrés, M. L. Barberis, I. Dotti, M. Fernández

备注：: 26 pages. Typos corrected
发表信息：: Osaka J. Math. 44, 765--793, 2007

英文摘要：

We study hermitian structures, with respect to the standard neutral metric on the cotangent bundle $T^*G$ of a 2n-dimensional Lie group $G$, which are left invariant with respect to the Lie group structure on $T^*G$ induced by the coadjoint action. These are in one-to-one correspondence with left invariant generalized complex structures on $G$. Using this correspondence and results of Cavalcanti-Gualtieri and Fernández-Gotay-Gray, it turns out that when $G$ is nilpotent and four or six dimensional, the cotangent bundle $T^*G$ always has a hermitian structure. However, we prove that if $G$ is a four dimensional solvable Lie group admitting neither complex nor symplectic structures, then $T^*G$ has no hermitian structure or, equivalently, $G$ has no left invariant generalized complex structure.

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

阅读原文

阅读 PDF ↗

arXiv 页面 ↗HTML 原文 ↗

本文内容

英文摘要相关论文

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

继续探索

相关论文

Natural Diagonal Riemannian Almost Product and Para-Hermitian Cotangent Bundles

We obtain the natural diagonal almost product and locally product structures on the total space of the cotangent bundle of a Riemannian manifold. We find the Riemannian almost product (locally product) and the (almost) para-Hermitian cotangent bundles of natural diagonal lift type. We prove the characterization theorem for the natural diagonal (almost) para-Kählerian structures on the total spaces of the cotangent bundle.

2011-04-14math.DG

Invariant hyperkahler structures on the cotangent bundles of Hermitian symmetric spaces

Let $G/K$ be an irreducible Hermitian symmetric spaces of compact type with the standard homogeneous complex structure. Then the real symplectic manifold $(T^*(G/K),Ω)$ has the natural complex structure $J^-$. We construct all $G$-invariant Kähler structures $(J,Ω)$ on homogeneous domains in $T^*(G/K)$ anticommuting with $J^-$. Each such a hypercomplex structure, together with a suitable metric, defines a hyperkähler structure. As an application, we obtain a new proof of the Harish-Chandra and Moore theorem.

2003-04-16math.CVmath.DG

Cotangent Bundles with General Natural Kahler Structures

We study the conditions under which an almost Hermitian structure $(G,J)$ of general natural lift type on the cotangent bundle $T^*M$ of a Riemannian manifold $(M,g)$ is K\" ahlerian. First, we obtain the algebraic conditions under which the manifold $(T^*M,G,J)$ is almost Hermitian. Next we get the integrability conditions for the almost complex structure $J$, then the conditions under which the associated 2-form is closed. The manifold $(T^*M,G,J)$ is K\" ahlerian iff it is almost Kahlerian and the almost complex structure $J$ is integrable. It follows that the family of Kahlerian structures…

2008-10-08math.DG

A class of Kaehler Einstein structures on the cotangent bundle

We use some natural lifts defined on the cotangent bundle T*M of a Riemannian manifold (M,g) in order to construct an almost Hermitian structure (G,J) of diagonal type. The obtained almost complex structure J on T*M is integrable if and only if the base manifold has constant sectional curvature and the coefficients as well as their derivatives, involved in its definition, do fulfill a certain algebraic relation. Next one obtains the condition that must be fulfilled in the case where the obtained almost Hermitian structure is almost Kaehlerian. Combining the obtained results we get a family of …

2004-05-14math.DG

A Kaehler Einstein structure on the cotangent bundle of a Riemannian manifold

We use the natural lifts of the fundamental tensor field g to the cotangent bundle T*M of a Riemannian manifold (M,g), in order to construct an almost Hermitian structure (G,J) of diagonal type on T*M. The obtained almost complex structure J on T*M is integrable if and only if the base manifold has constant sectional curvature and the second coefficient, involved in its definition is expressed as a rational function of the first coefficient and its first order derivative. Next one shows that the obtained almost Hermitian structure is almost Kaehlerian. Combining the obtained results we get a f…

2003-08-15math.DG

A canonical hyperkaehler metric on the total space of a cotangent bundle

A canonical hyperkaehler metric on the total space $T^*M$ of a cotangent bundle to a complex manifold $M$ has been constructed recently by the author (see alg-geom/9710026). This paper presents the results of alg-geom/9710026 in a streamlined and simplified form. The only new result is an explicit formula obtained for the case when $M$ is an Hermitian symmetric space.

2000-11-30math.AGmath.DG

更多相关论文收起更多论文

球面余切丛的上同调同构及其紧致化

Cohomology Isomorphism of Symmetric Power of Cotangent Bundle of Ball quotient and Its Toroidal Compactification

本文研究了有限体积复超曲面空间的对称幂余切丛的 $L^2$-Dolbeault 上同调及其紧致化，利用Hodge理论建立了Hodge分解和Green算子，并扩展了先前结果到有限体积空间。

2023-11-30math.CVmath.DG

平坦的Hermite李代数是Kähler的

Flat Hermitian Lie algebras are Kähler

Chongqing Normal University（重庆师范大学）

本文证明平坦的Hermite李代数必为Kähler，通过分析2步可解李群的Hermite几何性质，得出平坦Hermite结构的李群必为Kähler的结论。

2025-04-11math.DG

Invariant strong KT geometry on four-dimensional solvable Lie groups

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.

2009-11-03math.DG

Almost para-Hermitian and almost paracontact metric structures induced by natural Riemann extensions

In this paper we consider a manifold $(M,\nabla )$ with a symmetric linear connection $\nabla $ which induces on the cotangent bundle $T^*M$ of $M$ a semi-Riemannian metric $\overline g$ with a neutral signature. The metric $\overline g$ is called natural Riemann extension and it is a generalization (made by M. Sekizawa and O. Kowalski) of the Riemann extension, introduced by E. K. Patterson and A. G. Walker (1952). We construct two almost para-Hermitian structures on $(T^*M,\overline g)$ which are almost para-Kähler or para-Kähler and prove that the defined almost para-complex structures are …

2018-03-25math.DG

Complex Line Bundles over Simplicial Complexes and their Applications

Discrete vector bundles are important in Physics and recently found remarkable applications in Computer Graphics. This article approaches discrete bundles from the viewpoint of Discrete Differential Geometry, including a complete classification of discrete vector bundles over finite simplicial complexes. In particular, we obtain a discrete analogue of a theorem of André Weil on the classification of hermitian line bundles. Moreover, we associate to each discrete hermitian line bundle with curvature a unique piecewise-smooth hermitian line bundle of piecewise constant curvature. This is then us…

2017-01-18cs.DMmath.DG

Four dimensional symplectic Lie algebras

In this paper we deal with symplectic Lie algebras. All symplectic structures are determined for dimension four and the corresponding Lie algebras are classified up to equivalence. Symplectic four dimensional Lie algebras are described either as solutions of the cotangent extension problem or as symplectic double extension of $\Bbb RR^2$ by $\Bbb RR$. The difference in the choice of a certain model lies on the existence or not of a lagrangian ideal. Moreover all extensions of a two dimensional Lie algebra are determined, and so all solutions (up to equivalence) of the cotangent extension probl…

2004-07-28math.DG

关于余维数为2的阿贝尔理想的李代数的Streets-Tian猜想

Streets-Tian Conjecture on Lie algebras with codimension $2$ abelian ideals

本文针对李代数含余维数2阿贝尔理想的李复流形，通过显式构造度量与形变路径，验证了Streets-Tian猜想，拓展了该猜想在非凯勒几何中的适用情形。

2024-10-07math.DG

关于李群余切丛的Kähler量子化的BKS配对

On the BKS pairing for Kahler quantizations of the cotangent bundle of a Lie group

本文研究了紧致李群余切丛的Kähler量化，证明其BKS配对映射是酉的且等同于量子联络的平行移动，并揭示了其酉性源于Kähler极化退化时的Hermitian结构连续性。

2006-01-11math.DGmath.SG

关于广义度量结构

On generalized metric structures

本文研究了广义度量结构的可积条件，探讨了其与Courant括号和仿射连接诱导括号的可积性，以及与伪黎曼几何和广义Hermitian结构的关系。

2025-12-30math.DG

几乎阿贝尔李群与可解流形上的调和殆复结构

Harmonic almost complex structures on almost abelian Lie groups and solvmanifolds

该研究针对几乎阿贝尔李群，刻画了其上殆复结构关于埃尔米特度量调和的殆埃尔米特结构，适配了Gray-Hervella分类，并在相关紧可解流形上给出不同类的调和殆复结构实例。

2024-08-13math.DG

Chern numbers and the geometry of partial flag manifolds

We calculate the Chern classes and Chern numbers for the natural almost Hermitian structures of the partial flag manifolds F_n=SU(n+2)/S(U(n)\times U(1)\times U(1)). For all n>1 there are two invariant complex algebraic structures, which arise from the projectivizations of the holomorphic tangent and cotangent bundles of complex projective spaces. The projectivization of the cotangent bundle is the twistor space of a Grassmannian considered as a quaternionic Kähler manifold. There is also an invariant nearly Kähler structure, because F_n is a 3-symmetric space. We explain the relations between…

2008-02-27math.AGmath.DG

On the Chern-Ricci flow and its solitons for Lie groups

This paper is concerned with Chern-Ricci flow evolution of left-invariant hermitian structures on Lie groups. We study the behavior of a solution, as t is approaching the first time singularity, by rescaling in order to prevent collapsing and obtain convergence in the pointed (or Cheeger-Gromov) sense to a Chern-Ricci soliton. We give some results on the Chern-Ricci form and the Lie group structure of the pointed limit in terms of the starting hermitian metric and, as an application, we obtain a complete picture for the class of solvable Lie groups having a codimension one normal abelian subgr…

2013-11-04math.DG

Hermitian curvature flow on complex homogeneous manifolds

In this paper we study a version of the Hermitian curvature flow (HCF). We focus on complex homogeneous manifolds equipped with induced metrics. We prove that this finite-dimensional space of metrics is invariant under the HCF and write down the corresponding ODE on the space of Hermitian forms on the underlying Lie algebra. Using these computations we construct HCF-Einstein metrics on $G$-homogeneous manifolds, where $G$ is a complexification of a compact simple Lie group. We conjecture that under the HCF any induced metric on such a manifold pinches towards the HCF-Einstein metric. For a nil…

2017-06-21math.CVmath.DG

Complexified Hermitian Symmetric Spaces, Hyperkähler Structures, and Real Group Actions

There is a known hyperkähler structure on any complexified Hermitian symmetric space $G/K$, whose construction relies on identifying $G/K$ with both a (co)adjoint orbit and the cotangent bundle to the compact Hermitian symmetric space $G_u/K_0$. Via a family of explicit diffeomorphisms, we show that almost all of the complex structures are equivalent to the one on $G/K$; via a family of related diffeomorphisms, we show that almost all of the symplectic structures are equivalent to the one on $T^*\left(G_u/K_0\right)$. We highlight the intermediate Kähler structures, which share a holomorphic a…

2021-05-27math.DGmath.GRmath.SG

与余切丛相关的非分裂超流形

Non-split supermanifolds associated with the cotangent bundle

本文提出一种构造，将任意闭 (1,1)-形式关联到以分裂超流形为收缩核的非分裂超流形，并在不可约紧 Hermitian 对称空间情形下给出完全分类，同时计算了切层上同调，并研究了 $Π$-对称超格拉斯曼流形。

2022-12-13math-phmath.DGmath.MP

RC-positivity, vanishing theorems and rigidity of holomorphic maps

Let $M$ and $N$ be two compact complex manifolds. We show that if the tautological line bundle $\mathscr{O}_{T_M^*}(1)$ is not pseudo-effective and $\mathscr{O}_{T_N^*}(1)$ is nef, then there is no non-constant holomorphic map from $M$ to $N$. In particular, we prove that any holomorphic map from a compact complex manifold $M$ with RC-positive tangent bundle to a compact complex manifold $N$ with nef cotangent bundle must be a constant map. As an application, we obtain that there is no non-constant holomorphic map from a compact Hermitian manifold with positive holomorphic sectional curvature …

2018-09-04math.AGmath.CVmath.DG