Killing magnetic curves in non-flat Lorentzian-Heisenberg spaces

黎曼与洛伦兹 Killing 淹没中预定平均曲率图的二元性

A duality for prescribed mean curvature graphs in Riemannian and Lorentzian Killing submersions

本文针对黎曼与洛伦兹 Killing 淹没中的类空图建立共形二元性，交换平均曲率与丛曲率并互逆 Killing 长度，由此得到洛伦兹-闵可夫斯基空间中预定平均曲率完整图的存在性及与临界平均曲率相关的存在与不存在条件。

Imaginary Killing Spinors in Lorenztian Geometry

We study the geometric structure of Lorentzian spin manifolds, which admit imaginary Killing spinors. The discussion is based on the cone construction and a normal form classification of skew-adjoint operators in signature $(2,n-2)$. Derived geometries include Brinkmann spaces, Lorentzian Einstein-Sasaki spaces and certain warped product structures. Exceptional cases with decomposable holonomy of the cone are possible.

Conformal Killing spinors and special geometric structures in Lorentzian geometry - a survey

This paper is a survey on special geometric structures that admit conformal Killing spinors based on lectures, given at the ``Workshop on Special Geometric Structures in String Theory'', Bonn, September 2001 and at ESI, Wien, November 2001. We discuss the case of Lorentzian signature and explain which geometries occur up to dimension 6.

Explicit formulas for Killing magnetic curves in Heisenberg group

In this paper, We present the geometry three dimensional Heisenberg group (H_3,g) and its geodesics curves. After, we study the Killing magnetic curves and some geodesic Killing magnetic curves with its explicit formulas for such curves.

Conformally Einstein Lorentzian Lie groups with Heisenberg symmetry

We describe all Lorentzian semi-direct extensions of the Heisenberg group which are conformally Einstein. As a by side result, Bach-flat left-invariant Lorentzian metrics on semi-direct extensions of the Heisenberg group are classified, thus providing new background solutions in conformal gravity.

三维洛伦兹希格纳集团中时状极小曲面的奇点

Singularities on timelike minimal surfaces in Lorentzian Heisenberg group

研究了洛伦兹希格纳群中时状极小曲面的构造及其奇点特性，通过调和映射和具体例子探讨了不同奇点类型。

Constant angle surfaces in the Lorentzian Heisenberg group

In this paper, we define and, then, we characterize constant angle spacelike and timelike surfaces in the three-dimensional Heisenberg group, equipped with a 1-parameter family of Lorentzian metrics. In particular, we give an explicit local parametrization of these surfaces and we produce some examples.

洛伦兹海森堡群中的螺旋曲面

Helix surfaces in Lorentzian Heisenberg group

本文在洛伦兹海森堡群中完整分类了极小与常平均曲率螺旋曲面并给出显式参数化，同时刻画了此前未研究的常角类空与类时曲面，证明其具有常高斯曲率。

联络的像与洛伦兹局部对称空间的卡拉比算子

The range of a connection and a Calabi operator for Lorentzian locally symmetric spaces

本文扩展黎曼情形结果，确定洛伦兹局部对称空间中卡拉比算子可识别基灵算子像的情况，不可分解空间均满足，精确找出不满足的乘积空间，方法是先建立联络像的判别准则再应用于基灵联络。

Lorentzian Heisenberg群中零均曲率表面的对偶性

Duality of zero mean curvature surfaces in the Lorentzian Heisenberg group

研究Heisenberg群中零均曲率表面的对偶变换，并证明其具有零均曲率，推导出双曲面的Sym公式。

Hyperbolic evolution equations, Lorentzian holonomy, and Riemannian generalised Killing spinors

We prove that the Cauchy problem for parallel null vector fields on smooth Lorentzian manifolds is well posed. The proof is based on the derivation and analysis of suitable hyperbolic evolution equations given in terms of the Ricci tensor and other geometric objects. Moreover, we classify Riemannian manifolds satisfying the constraint conditions for this Cauchy problem. It is then possible to characterise certain holonomy reductions of globally hyperbolic manifolds with parallel null vector in terms of flow equations for Riemannian special holonomy metrics. For exceptional holonomy groups thes…

Extensions maximales et classification des tores lorentziens munis d'un champ de Killing

We study the simply connected inextendable Lorentzian surfaces admitting a Killing vector field. We construct a natural family of such surfaces, that we call "universal extensions". They are characterized by a condition of symmetry, the "reflexivity", and a by a rather weak completeness assumption, the absence of "saddles at infinity". Considering these surfaces as model spaces, we study their minimal quotients, divisible open sets and conjugate points. We show uniformisation results (by an open subset of one of these universal extensions, which is uniquely determined) in the following cases: …

黎曼与洛伦兹三维流形上的 Killing 向量场

Killing vector fields on Riemannian and Lorentzian 3-manifolds

本文完整局部分类了具有非零Killing向量场的黎曼与洛伦兹三维流形，推广了Sasakian结构，并给出局部共形平坦及全局完备性和常正曲率度量的条件。

Killing superalgebras for Lorentzian four-manifolds

We determine the Killing superalgebras underpinning field theories with rigid unextended supersymmetry on Lorentzian four-manifolds by re-interpreting them as filtered deformations of $\mathbb{Z}$-graded subalgebras with maximum odd dimension of the $N{=}1$ Poincaré superalgebra in four dimensions. Part of this calculation involves computing a Spencer cohomology group which, by analogy with a similar result in eleven dimensions, prescribes a notion of Killing spinor, which we identify with the defining condition for bosonic supersymmetric backgrounds of minimal off-shell supergravity in four d…

海森堡群上的洛伦兹代数 Ricci 孤立子

Lorentzian Algebraic Ricci Solitons on the Heisenberg Group

本文研究了海森堡群上三种洛伦兹度量对应的代数 Ricci 孤立子，确定了 $\mu$ 为收缩型、$\nu$ 仅在三维时收缩、$\phi$ 在 $n>1$ 时为稳态，$n=1$ 时平坦。

Killing superalgebras for Lorentzian six-manifolds

We calculate the Spencer cohomology of the $(1,0)$ Poincaré superalgebras in six dimensions: with and without R-symmetry. As the cases of four and eleven dimensions taught us, we may read off from this calculation a Killing spinor equation which allows the determination of which geometries admit rigidly supersymmetric theories in this dimension. We prove that the resulting Killing spinors generate a Lie superalgebra and determine the geometries admitting the maximal number of such Killing spinors. They are divided in two branches. One branch consists of the lorentzian Lie groups with bi-invari…

Killing superalgebras for lorentzian five-manifolds

We calculate the relevant Spencer cohomology of the minimal Poincaré superalgebra in 5 spacetime dimensions and use it to define Killing spinors via a connection on the spinor bundle of a 5-dimensional lorentzian spin manifold. We give a definition of bosonic backgrounds in terms of this data. By imposing constraints on the curvature of the spinor connection, we recover the field equations of minimal (ungauged) 5-dimensional supergravity, but also find a set of field equations for an $\mathfrak{sp}(1)$-valued one-form which we interpret as the bosonic data of a class of rigid supersymmetric th…

次洛伦兹Heisenberg群上的最优传输

Optimal transport on the sub-Lorentzian Heisenberg group

本文研究了次洛伦兹Heisenberg群的合成度量时空结构及最优传输问题，建立了次洛伦兹版本的Brenier定理，并给出了最优传输映射的例子及推导出了次洛伦兹Monge-Ampère方程。

Pseudo-Riemannian Symmetries on Heisenberg group $\mathbb{H}_{3}$

The notion of $Γ$-symmetric space is a natural generalization of the classical notion of symmetric space based on $\z_2$-grading of Lie algebras. In our case, we consider homogeneous spaces $G/H$ such that the Lie algebra $\g$ of $G$ admits a $Γ$-grading where $Γ$ is a finite abelian group. In this work we study Riemannian metrics and Lorentzian metrics on the Heisenberg group $\mathbb{H}_3$ adapted to the symmetries of a $Γ$-symmetric structure on $\mathbb{H}_3$. We prove that the classification of $\z_2^2$-symmetric Riemannian and Lorentzian metrics on $\mathbb{H}_3$ corresponds to the class…

洛伦兹海森堡群中的极大曲面

Maximal surfaces in the Lorentzian Heisenberg group

本文研究洛伦兹海森堡群中平均曲率为零的类空曲面，通过调和映射和圈群方法刻画其一般奇点类型，并证明类光边界正则极大圆盘至少含两个尖交叉帽奇点。

海森堡群上的洛伦兹问题

Lorentzian problems on the Heisenberg group

针对海森堡群上的三个左不变洛伦兹问题，通过庞特里亚金最大值原理获得异常与正常极值轨迹的参数化，同时研究了可达集及最优轨迹的存在性。

子洛伦兹海森堡群中的常平均曲率曲面

Constant mean curvature surfaces in the sub-Lorentzian Heisenberg group

本文研究了子洛伦兹海森堡群中具有常水平平均曲率的曲面，通过体积保持径向变分推导了水平面积的第一变分公式，并完整分类了光滑的boost对称常平均曲率曲面，发现了一族非零常平均曲率的光滑非因果boost对称曲面，推测其为等周问题的解。

A classification of left-invariant Lorentzian metrics on some nilpotent Lie groups

It has been known that there exist exactly three left-invariant Lorentzian metrics up to scaling and automorphisms on the three dimensional Heisenberg group. In this paper, we classify left-invariant Lorentzian metrics on the direct product of three dimensional Heisenberg group and the Euclidean space of dimension $n-3$ with $n \geq 4$, and prove that there exist exactly six such metrics on this Lie group up to scaling and automorphisms. Moreover we show that only one of them is flat, and the other five metrics are Ricci solitons but not Einstein. We also characterize this flat metric as the u…

Charged Conformal Killing Spinors

We study the twistor equation on pseudo-Riemannian $Spin^c-$manifolds whose solutions we call charged conformal Killing spinors (CCKS). We derive several integrability conditions for the existence of CCKS and study their relations to spinor bilinears. A construction principle for Lorentzian manifolds admitting CCKS with nontrivial charge starting from CR-geometry is presented. We obtain a partial classification result in the Lorentzian case under the additional assumption that the associated Dirac current is normal conformal and complete the Classification of manifolds admitting CCKS in all di…

On the scalar curvature of hypersurfaces in spaces with a Killing field

We consider compact hypersurfaces in an $(n+1)$-dimensional either Riemannian or Lorentzian space $N^{n+1}$ endowed with a conformal Killing vector field. For such hypersurfaces, we establish an integral formula which, especially in the simpler case when $N=M^n\times R$ is a product space, allows us to derive some interesting consequences in terms of the scalar curvature of the hypersurface. For instance, when $n=2$ and $M^2$ is either the sphere $\mathbb{S}^2$ or the real projective plane $\mathbb{RP}^2$, we characterize the slices of the trivial totally geodesic foliation $M^2\times\{t\}$ as…

Examples and structure of CMC surfaces in some Riemannian and Lorentzian homogeneous spaces

It is proved that the holomorphic quadratic differential associated to CMC surfaces in Riemannian products $\mathbb{S}^2\times\Rr$ and $\mathbb{H}^2\times \Rr$ discovered by U. Abresch and H. Rosenberg could be obtained as a linear combination of usual Hopf differentials. Using this fact, we are able to extend it for Lorentzian products. Families of examples of helicoidal CMC surfaces on these spaces are explicitly described. We also present some characterizations of CMC rotationally invariant discs and spheres. Finally, after establish some height and area estimates, we prove the existence of…

亚洛伦兹海森堡群中的豪斯多夫维数与合成曲率界的失效

Hausdorff dimension and failure of synthetic curvature bounds in the sub-Lorentzian Heisenberg group

本文通过初等变分法证明亚洛伦兹海森堡群的极大测地线为双曲线提升，确定其洛伦兹豪斯多夫维数为4，并揭示该群不满足任何参数的类时曲率维数条件与测度收缩性质。

$\mathcal{I}$-Preserving Diffeomorphisms of Lorentzian Manifolds

We examine the existence of one parameter groups of diffeomorphisms whose infinitesimal generators annihilate all scalar polynomial curvature invariants through the application of the Lie derivative, known as $\mathcal{I}$-preserving diffeomorphisms. Such mappings are a generalization of isometries and appear to be related to nil-Killing vector fields, for which the associated Lie derivative of the metric yields a nilpotent rank two tensor. We show that the set of nil-Killing vector fields contains Lie algebras, although the Lie algebras may be infinite and can contain elements which are not $…

Spinors of real type as polyforms and the generalized Killing equation

We develop a new framework for the study of generalized Killing spinors, where generalized Killing spinor equations, possibly with constraints, can be formulated equivalently as systems of partial differential equations for a polyform satisfying algebraic relations in the Kähler-Atiyah bundle constructed by quantizing the exterior algebra bundle of the underlying manifold. At the core of this framework lies the characterization, which we develop in detail, of the image of the spinor squaring map of an irreducible Clifford module $Σ$ of real type as a real algebraic variety in the Kähler-Atiyah…

Pseudo-Riemannian Symmetries on Heisenberg groups

The notion of $Γ$-symmetric space is a natural generalization of the classical notion of symmetric space based on $\Z_2$-grading of Lie algebras. In our case, we consider homogeneous spaces $G/H$ such that the Lie algebra $\g$ of $G$ admits a $Γ$-grading where $Γ$ is a finite abelian group. In this work we study Riemannian metrics and Lorentzian metrics on the Heisenberg group $\mathbb{H}_3$ adapted to the symmetries of a $Γ$-symmetric structure on $\mathbb{H}_3$. We prove that the classification of $\z$-symmetric Riemannian and Lorentzian metrics on $\mathbb{H}_3$ corresponds to the classific…

On the Geometry of Circle Bundles with Special Holonomy

We investigate geometric properties of indecomposable but non-irreducible Lorentzian manifolds, which are total spaces of circle bundles. We investigate under which conditions these manifolds are complete and give examples which fulfill the obtained conditions. In particular we investigate the Einstein equation for these spaces yielding examples for complete compact Ricci flat Lorentzian manifolds and manifolds with timelike Killing vector fields. Finally we study their holonomy and obtain in particular complete examples for Lorentzian manifolds with holonomy of so called type 4.

Geodesic completeness of some Lorentzian simple Lie groups

In this paper we investigate geodesic completeness of left-invariant Lorentzian metrics on a simple Lie group $G$ when there exists a left-invariant Killing vector field $Z$ on $G$. Among other results, it is proved that if $Z$ is timelike, or $G$ is strongly causal and $Z$ is lightlike, then the metric is complete. We then consider the special complex Lie group $SL_2(\mathbb{C})$ in more details and show that the existence of a lightlike vector field $Z$ on it, implies geodesic completeness. We also consider the existence of a spacelike vector field $Z$ on $SL_2(\mathbb{C})$ and provide an eq…

三维海森堡群中的极小零滚动面

Minimal null scrolls in the three-dimensional Heisenberg group

本文在三维海森堡群的左不变洛伦兹度量下，将一类满足Abresch-Rosenberg微分乘积消失条件的类时极小曲面刻画为零曲线与仿射零直线的乘积，并给出相应构造方法。

3-维流形上非紧致域的极小图

Minimal graphs over non-compact domains in 3-manifolds fibered by a Killing vector field

研究3维流形上非紧致域的极小Killing图，解决Dirichlet问题并证明唯一性和奇点可移除性。

基于洛伦兹调和映射的德西特2空间中的框架零曲面与类时间曲面

Framed null curves and timelike surfaces via Lorentzian harmonic maps into de-Sitter 2-space

本文基于框架零曲线构造了满足特定特征值方程的洛伦兹调和映射，并研究了两类类时间曲面的几何性质与奇异性特征。

局部共形同构的洛伦兹空间

Locally conformally homogeneous Lorentzian spaces

该研究证明了局部共形同构的洛伦兹流形要么共形平坦，要么局部等价于齐性平面波，并揭示了平面波结构与Penrose极限的关系。

二维 Killing 超代数

Killing Superalgebras in 2 Dimensions

本文在二维伪 Riemann 流形上研究非手征平坦超代数的 Spencer 上同调与滤过形变，证明其对应于不同号差下几何 Killing 及斜 Killing 旋量的 Killing 超代数。

Dual quadratic differentials and entire minimal graphs in Heisenberg space

We define holomorphic quadratic differentials for spacelike surfaces with constant mean curvature in the Lorentzian homogeneous spaces $\mathbb{L}(κ,τ)$ with isometry group of dimension 4, which are dual to the Abresch-Rosenberg differentials in the Riemannian counterparts $\mathbb{E}(κ,τ)$, and obtain some consequences. On the one hand, we give a very short proof of the Bernstein problem in Heisenberg space, and provide a geometric description of the family of entire graphs sharing the same differential in terms of a 2-parameter conformal deformation. On the other hand, we prove that entire m…

Isometry groups of Lorentzian manifolds of finite volume and The local geometry of compact homogeneous Lorentz spaces

Based on the work of Adams and Stuck as well as on the work of Zeghib, we classify the Lie groups which can act isometrically and locally effectively on Lorentzian manifolds of finite volume. In the case that the corresponding Lie algebra contains a direct summand isomorphic to the two-dimensional special linear algebra or to a twisted Heisenberg algebra, we also describe the geometric structure of the manifolds if they are compact. Using these results, we investigate the local geometry of compact homogeneous Lorentz spaces whose isometry groups have non-compact connected components. It turns …

Heisenberg空间$\nil$中不变Gauss曲率孤子的分类

Classification of invariant Gauss curvature solitons in the Heisenberg space $\nil$

本文分类了三维Heisenberg群$\mathrm{Nil}_3$中Gauss曲率流的孤子，这些孤子在一参数等距群下不变，通过四种Killing向量场和三族不变曲面分析了十二种可能的孤子类型，给出了显式参数化或几何性质。

磁 Killing 张量与磁流的首个积分

Magnetic Killing tensors and first integrals of the magnetic flow

本文提出磁 Killing 对称张量，用于构造磁流的首个积分，并证明某些 nilmanifolds 上磁流的可积性。

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…

Ricci flat left invariant Lorentzian metrics on 2-step nilpotent Lie groups

We determine all Ricci flat left invariant Lorentzian metrics on simply connected 2-step nilpotent Lie groups. We show that the $2k+1$-dimensional Heisenberg Lie group $H_{2k+1}$ carries a Ricci flat left invariant Lorentzian metric if and only if $k=1$. We show also that for any $2\leq q\leq k$, $H_{2k+1}$ carries a Ricci flat left invariant pseudo-Riemannian metric of signature $(q,2k+1-q)$ and we give explicite examples of such metrics.

The Dirichlet problem for constant mean curvature surfaces in Heisenberg space

We study constant mean curvature graphs in the Riemannian 3-dimensional Heisenberg spaces ${\cal H}={\cal H}(τ)$. Each such ${\cal H}$ is the total space of a Riemannian submersion onto the Euclidean plane $\mathbb{R}^2$ with geodesic fibers the orbits of a Killing field. We prove the existence and uniqueness of CMC graphs in ${\cal H}$ with respect to the Riemannian submersion over certain domains $Ω\subset\mathbb{R}^2$ taking on prescribed boundary values.

共形Killing--Yano Ricci孤立子：结构、相容性与刚性

Conformal Killing--Yano Ricci solitons: Structure, compatibility, and rigidity

本文引入共形Killing--Yano Ricci孤立子结构，建立相容性恒等式，证明在4维Lorentzian下其强制局部Kerr--NUT--(A)dS几何，并给出谱阻碍与显式构造，为研究隐藏对称性的共形Ricci孤立子提供几何框架。

Sasakian Lorentzian 3-维 manifold 中的接触涡线

Contact Whirl Curves in Sasakian Lorentzian 3-Manifolds

本文研究了Sasakian Lorentzian 3-维 manifold 中的接触涡线，通过Reeb向量场编码曲线适应框架与环境接触结构的相互作用，推导了非测地单位速度接触涡线的微分方程，并证明了非测地Legendre Frenet曲线具有常数 torsion τ=1。

Locally Homogeneous Spaces, Induced Killing Vector Fields and Applications to Bianchi Prototypes

An answer to the question: Can, in general, the adoption of a given symmetry induce a further symmetry, which might be hidden at a first level? has been attempted in the context of differential geometry of locally homogeneous spaces. Based on E. Cartan's theory of moving frames, a methodology for finding all symmetries for any n dimensional locally homogeneous space is provided. The analysis is applied to 3 dimensional spaces, whereby the embedding of them into a 4 dimensional Lorentzian manifold is examined and special solutions to Einstein's field equations are recovered. The analysis is mai…

Geometry and symmetries of null G-structures

We present a definition of null G-structures on Lorentzian manifolds and investigate their geometric properties. This definition includes the Robinson structure on 4-dimensional black holes as well as the null structures that appear in all supersymmetric solutions of supergravity theories. We also identify the induced geometry on some null hypersurfaces and that on the orbit spaces of null geodesic congruences in such Lorentzian manifolds. We give the algebra of diffeomorphisms that preserves a null G-structure and demonstrate that in some cases it interpolates between the BMS algebra of an as…

关于齐次平面波

On homogeneous plane waves

本文严格处理齐次平面波的数学性质，通过上齐次一海森堡作用将其视为闵可夫斯基时空的形变，确定了等距群的单位连通分量，并证明了非平坦齐次平面波的$C^2$-不可延展性、测地完备性条件及全局Brinkmann坐标的存在性。

Generalized Cylinders in Semi-Riemannian and Spin Geometry

We use a construction which we call generalized cylinders to give a new proof of the fundamental theorem of hypersurface theory. It has the advantage of being very simple and the result directly extends to semi-Riemannian manifolds and to embeddings into spaces of constant curvature. We also give a new way to identify spinors for different metrics and to derive the variation formula for the Dirac operator. Moreover, we show that generalized Killing spinors for Codazzi tensors are restrictions of parallel spinors. Finally, we study the space of Lorentzian metrics and give a criterion when two L…

Sub-Lorentzian Geometry on Anti-De Sitter Space

Sub-Riemannian Geometry is proved to play an important role in many applications, e.g., Mathematical Physics and Control Theory. The simplest example of sub-Riemannian structure is provided by the 3-D Heisenberg group. Sub-Riemannian Geometry enjoys major differences from the Riemannian being a generalisation of the latter at the same time, e.g., geodesics are not unique and may be singular, the Hausdorff dimension is larger than the manifold topological dimension. There exists a large amount of literature developing sub-Riemannian Geometry. However, very few is known about its natural extensi…

Contact integral geometry and the Heisenberg algebra

Generalizing Weyl's tube formula and building on Chern's work, Alesker reinterpreted the Lipschitz-Killing curvature integrals as a family of valuations (finitely-additive measures with good analytic properties), attached canonically to any Riemannian manifold, which is universal with respect to isometric embeddings. In this note, we uncover a similar structure for contact manifolds. Namely, we show that a contact manifold admits a canonical family of generalized valuations, which are universal under contact embeddings. Those valuations assign numerical invariants to even-dimensional submanifo…

Some criteria for Wind Riemannian completeness and existence of Cauchy hypersurfaces

Recently, a link between Lorentzian and Finslerian Geometries has been carried out, leading to the notion of wind Riemannian structure (WRS), a generalization of Finslerian Randers metrics. Here, we further develop this notion and its applications to spacetimes, by introducing some characterizations and criteria for the completeness of WRS's. As an application, we consider a general class of spacetimes admitting a time function $t$ generated by the flow of a complete Killing vector field (generalized standard stationary spacetimes or, more precisely, SSTK ones) and derive simple criteria ensur…