Khadidja Derkaoui, Fouzi Hathout
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.
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We obtain some explicit formulas for Killing magnetic curves in non-flat Lorentzian-Heisenberg spaces.
We consider the three-dimensional Heisenberg group, equipped with any left-invariant metric, either Lorentzian or Riemannian. We completely classify their affine vector fields and investigate their relationship with Killing vector fields and their casual character. We also classify their Ricci, curvature and matter collineations, proving that there are no nontrivial examples.
We study the Minkowski formula of conformal Killing-Yano two-forms in a spacetime of constant curvature. We establish the spacetime Alexandrov theorem with a free boundary.
In this article, we will discuss the smooth $(X_{M}+\sqrt{-1}Y_{M})$-invariant forms on $M$ and to establish a localization formulas. As an application, we get a localization formulas for characteristic numbers.
Conformal Killing forms on $2$-step nilpotent Riemannian Lie groups
该研究针对单连通2步幂零黎曼李群的左不变共形Killing 2、3-形式展开，证明中心维数≥4时这类形式均为Killing形式，还刻画了中心维数≤3时存在非上闭形式的所有群及其形式空间。
Killing forms on G2 and Spin7 manifolds
黎曼流形上的Killing形式是其共变导数是全反对称的微分形式。我们证明了，在一个紧致流形与holonomy G2或Spin7任何Killing形式必须是平行的。主要工具是一个通用的Weitzenboeck公式。我们展示了这样一个公式可以得到任何给定的和乐群和任何表示定义一个向量丛。
Killing frames and S-curvature of homogeneous Finsler spaces
本文首先利用Killing向量场导出了齐次Finsler空间的S-曲率公式。证明了齐次Finsler空间具有迷向S-曲率当且仅当它具有消失S-曲率。在齐次Finsler空间是Randers空间的特殊情况下，我们给出了一个显式公式，它与第二作者用其他方法得到的公式一致。
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。
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。
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…
Tachibana, Killing and planarity numbers of compact Riemannian manifolds
本文给出了黎曼流形上的共形Killing数、Killing数和平面型的定义和性质，并确定了Tachibana数、Killing数和平面型数作为著名的贝蒂数的类似数.我们陈述一些条件来描述这些数。此外，我们制定的Betti，Tachibana，Killing和平面数之间的关系的主要结果。
Invariant $λ$-translators in the Heisenberg group
研究Heisenberg空间$\mathrm{Nil}_3$中平均曲率$H=\langle N,\partial_z\rangle+\lambda$的定向曲面，分类在左平移、旋转和螺旋运动下不变的$\lambda$-平移子。
In this note, we introduce a new type of warped products called as sequential warped products to cover a wider variety of exact solutions to Einstein's equation. First, we study the geometry of sequential warped products and obtain covariant derivatives, curvature tensor, Ricci curvature and scalar curvature formulas. Then some important consequences of these formulas are also stated. We provide characterizations of geodesics and two different types of conformal vector fields, namely, Killing vector fields and concircular vector fields on sequential warped product manifolds. Finally, we consid…
In this article, we will discuss a localization formulas of equivariant cohomology about two Killing vector fields on the set of zero points ${\rm{Zero}}(X_{M}-\sqrt{-1}Y_{M})=\{x\in M \mid |Y_{M}(x)|=|X_{M}(x)|=0 \}.$ As application, we use it to get formulas about characteristic numbers and to get a Duistermaat-Heckman type formula on symplectic manifold.
Twelve characterisations of Killing vector fields in Finsler geometry
本文在Finsler几何中利用Frölicher-Nijenhuis形式体系，建立了Killing向量场的十二种等价刻画，并给出局部方程形式，揭示其多方面的几何本质。
Sym-Bobenko formula for minimal surfaces in Heisenberg space
给出了三维Heisenberg空间中极小曲面的浸入公式，即Sym-Bobenko公式.这样的公式可以用来给出广义Weierstrass型表示和构造极小曲面的显式例子。
An integral formula on the Heisenberg group
设Hn表示（2n + 1）维（亚黎曼）海森堡群.本文将证明一个积分恒等式（见定理1.2），它推广了70年代Reilly得到的一个公式。第一部分的应用将在第4节中给出。
Euler characteristic and Lipschitz-Killing curvatures of closed semi-algebraic sets
证明了一个将闭半代数集的Euler-Poincaré特征与Lipschitz-Killing曲率联系起来的公式
Explicit Formulas for Non-Geodesic Biharmonic Curves of the Heisenberg Group
本文研究了黎曼流形间映射的双调和条件（见[BK]），并研究了Heisenberg群H_3中的非测地双调和曲线。首先证明它们都是螺旋，然后明确地得到它们的参数方程。
We prove an almost monotonicity formula for H-minimal Legendrian Surfaces (also called {\it contact stationary legendrian immersions}) in the Heisenberg Group ${\mathbb H}^2$ . From this formula we deduce a Bernstein-Liouville type theorem for H-minimal Legendrian Surfaces. This last result happens to be in particular a central ingredient in the analysis part of the proof of the Willmore conjecture given by the author in arXiv:2007.05467
Spinor equations in Weyl geometry
本文研究了具有CSpin结构的Weyl流形上的Dirac方程、扭量方程和Killing方程。给出了一个共形的Schr“odinger-Lichnerowicz公式，并利用它给出了这些方程的可积性条件。通过引入任意权旋量的Killing方程，将AndreiMoroianu在[9]中的结果推广到如下意义.唯一的非封闭Weyl流形的维数大于大于3，承认解决方案的真正Killing方程是4维和非紧。任何在这些维中存在一个真正的Killing旋量的外尔流形都必须是爱因斯坦-外尔流形。
Coupling solutions of BGG-equations in conformal spin geometry
BGG-方程是抛物几何上偏微分方程的几何超定组。BGG-方程的正规解是特别有趣的，我们给出了一个简单的公式的必要和充分的额外的可积性条件的解决方案。然后，我们讨论了一个程序耦合已知的解决方案的BGG方程产生新的。采用适当的演算共形自旋结构，这产生明确的耦合公式和条件之间的几乎爱因斯坦尺度，共形Killing形式和扭量旋量。最后，我们讨论了一类通用扭量旋量，提供了一个不变的共形Killing场的分解。
Irreducible Killing and conformal Killing tensors on homogeneous plane waves
该研究借助 BGG 算子分类齐性平面波上的不可约 Killing 与共形 Killing 2-张量，并给出显式公式及存在条件。
We show the analogy of Cauchy's surface area formula for the Heisenberg groups $\mathbb{H}_n$ for $n\geq 1$, which states that the p-area of any compact hypersurface $Σ$ in $\mathbb{H}_n$ with its p-normal vector defined almost everywhere on $Σ$ is the average of its projected p-areas onto the orthogonal complements of all p-normal vectors of the Pansu spheres (up to a constant). The formula provides a geometric interpretation of the p-areas defined by Cheng-Hwang-Malchiodi-Yang [9] in $\mathbb{H}_1$ and Cheng-Hwang-Yang [7] in $\mathbb{H}_n$ for $n\geq 2$. We also characterize the projected a…
By studying the group of rigid motions, $PSH(1)$, in the 3D-Heisenberg group $H_1$, we define the density and the measure for the sets of horizontal lines. We show that the volume of a convex domain $D\subset H_1$ is equal to the integral of length of chord over all horizontal lines intersecting $D$. As the classical result in integral geometry, we also define the kinematic density for $PSH(1)$ and show the probability of randomly throwing a vector $v$ interesting the convex domain $D\subset D_0$ under the condition that $v$ is contained in $D_0$. Both results show the relationship connecting …
The recently introduced Lipschitz-Killing curvature measures on pseudo-Riemannian manifolds satisfy a Weyl principle, i.e. are invariant under isometric embeddings. We show that they are uniquely characterized by this property. We apply this characterization to prove a Künneth-type formula for Lipschitz-Killing curvature measures, and to classify the invariant generalized valuations and curvature measures on all isotropic pseudo-Riemannian space forms.
An index formula for the extended Heisenberg algebra of Epstein, Melrose and Mendoza
接触流形的扩展海森堡代数有一个符号演算，它既包含海森堡伪微分算子，也包含经典的伪微分算子。本文导出了在这种推广的微积分中弗雷德霍姆算子的指数公式。这个公式包含在一个单一的表达的阿蒂亚-辛格公式的椭圆算子，以及布泰德蒙维尔的Toeplitz指数公式。
Monotonicity of Causal Killing Vectors and Geometry of ADM Mass Minimizers
本文研究了ADM质量极小化初始数据集，证明了正质量定理等号情形与正Bartnik质量极小化子的时空嵌入性质，并利用因果Killing向量场Lorentzian长度的新单调性公式证实了Bartnik平稳真空猜想。
Killing spinor equations in dimension 7 and geometry of integrable G_2-manifolds
我们计算了7维${G}_2$ -流形的数量曲率，其中流形允许一个具有全反对称挠率的联系。本文证明了Killing旋量方程的通解公式，并用伸缩函数和NS 3型场表示了解的黎曼数量曲率.在n=7维中，第二费米子弦方程中的膨胀函数可以解释为基本的可积${G}_2$ -结构共形变化为纯$W_3$型的共形变化。
Killing tensor fields have been thought of as describing hidden symmetry of space(-time) since they are in one-to-one correspondence with polynomial first integrals of geodesic equations. Many problems in classical mechanics can be formulated as geodesic problems in curved spaces and spacetimes, and thus solving the defining equation for Killing tensor fields (the Killing equation) is a powerful way to integrate the equations of motion. In this paper we attempt to formulate the integrability conditions of the Killing equation, which serve to put an upper bound on the number of linearly indepen…
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…
In the present paper, we obtain explicit formulae for geodesics in some left-invariant sub-Finsler problems on Heisenberg groups $\mathbb{H}_{2n+1}$. Our main assumption is the following: the compact convex set of unit velocities at identity admits a generalization of spherical coordinates. This includes convex hulls and sums of coordinate 2-dimensional sets, all left-invariant sub-Riemannian structures on $\mathbb{H}_{2n+1}$, and unit balls in $L_p$-metric for $1\le p\le\infty$. In the last case, extremals are obtained in terms of incomplete Euler integral of the first kind.
Blow-up of regular submanifolds in Heisenberg groups and applications
在Heisenberg群中，利用内禀膨胀，得到了正则子流形的爆破定理。这个结果的主要结果是一个关于黎曼曲面测度的p维子流形的（p+1）维球面Hausdorff测度的密度的显式公式。我们在一些简单的例子中明确地计算了这个公式，并给出了关于流的弱收敛的球形Hausdorff测度的下半连续性结果。另一个应用是证明海森堡群上向量值映射的一个内禀余面积公式。
We explore the geometry that underlies the osculating nilpotent group structures of the Heisenberg calculus. For a smooth manifold $M$ with a distribution $H\subseteq TM$ analysts use explicit (and rather complicated) coordinate formulas to define the nilpotent groups that are central to the calculus. Our aim in this paper is to provide insight in the intrinsic geometry that underlies these coordinate formulas. First, we introduce `parabolic arrows' as a generalization of tangent vectors. The definition of parabolic arrows involves a mix of first and second order derivatives. Parabolic arrows …
Determinant of Laplacians on Heisenberg Manifolds
本文给出了三维Heisenberg流形上Laplacian的zeta正则化行列式的一个积分表示，并研究了当一致离散子群变形时其值的变化. Heiseberg流形是一个纤维丛的全空间，它以环面为底空间，以圆为典型纤维，那么均匀离散子群的变形意味着纤维的“半径”趋于零。我们精确地解释了三维情形的计算结果，并给出了五维海森堡流形的相应结果。我们看到，这些值本身具有乘积形式，其因子是平坦环面的因子。因此，本文后半部分给出了两个黎曼流形的乘积型流形的ζ正则化行列式的一般公式，讨论了平坦环面的公式，并说明了二维平坦环面的公式与Kronecker第二极限公式的关系。
A note on traces for the Heisenberg calculus
本文针对先前工作中接触流形上海森堡椭圆算子指标公式里的关键新迹，阐明了其在海森堡伪微分算子代数上的性质，该迹曾是构造周期循环上同调上循环以计算指标的复杂成分。
On Sasakian quasi-Killing spinors in three-dimensions
本文研究三维伪 Riemann Sasakian 空间形式上的 SqK 旋量，揭示其与 Reeb 场、带电粒子运动及 Einstein-Dirac-Maxwell 系统的联系，并给出显式公式。
We strongly develop the relationship between complex hyperbolic geometry and arithmetic counting or equidistribution applications, that arises from the action of arithmetic groups on complex hyperbolic spaces, especially in dimension $2$. We prove a Mertens' formula for the integer points over a quadratic imaginary number fields $K$ in the light cone of Hermitian forms, as well as an equidistribution theorem of the set of rational points over $K$ in Heisenberg groups. We give a counting formula for the cubic points over $K$ in the complex projective plane whose Galois conjugates are orthogonal…
We introduce novel equations, in the spirit of rough path theory, that parametrize level sets of intrinsically regular maps on the Heisenberg group with values in $\mathbb{R}^2$. These equations can be seen as a sub-Riemannian counterpart to classical ODEs arising from the implicit function theorem. We show that they enjoy all the natural well-posedness properties, thus allowing for a "good calculus" on nonsmooth level sets. We apply these results to prove an area formula for the intrinsic measure of level sets, along with the corresponding coarea formula.
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