R. Caddeo, C. Oniciuc, P. Piu
We consider the biharmonicity condition for maps between Riemannian manifolds (see [BK]), and study the non-geodesic biharmonic curves in the Heisenberg group H_3. First we prove that all of them are helices, and then we obtain explicitly their parametric equations.
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
In this paper we study the non-geodesic non-null biharmonic curves in 3-dimensional hyperbolic Heisenberg group. We prove that all of the non-geodesic non-null biharmonic curves in 3-dimensional hyperbolic Heisenberg group are helices. Moreover, we obtain explicit parametric equations for non-geodesic non-null biharmonic curves and non-geodesic spacelike horizontal biharmonic curves in 3-dimensional hyperbolic Heisenberg group, respectively. We also show that there do not exist non-geodesic timelike horizontal biharmonic curves in 3-dimensional hyperbolic Heisenberg group.
Biharmonic curves are a generalization of geodesics, with applications in elasticity theory and various branches of computer science. The paper proposes a first study of biharmonic curves in spaces with Finslerian geometry, covering the following topics: a deduction of their equations, existence of non-geodesic biharmonic curves for some classes of Finsler spaces and specific properties. We prove, for instance, that geodesic curvature is constant along Finslerian biharmonic curves. Integration of the biharmonic equation is presented for two concrete Finsler metrics.
Sasakian manifolds provide explicit formulae of some Jacobi operators which describe the biharmonic equation of curves in Riemannian manifolds. In this paper we characterize non-geodesic biharmonic curves in Sasakian manifolds which are either tangent or normal to the Reeb vector field. In the three-dimensional case, we prove that such curves are some helixes whose geodesic curvature and geodesic torsion satisfy a given relation.
In this paper, we study biharmonic maps into Sol and Nil spaces, two model spaces of Thurston's 3-dimensional geometries. We characterize non-geodesic biharmonic curves in Sol space and prove that there exists no non-geodesic biharmonic helix in Sol space. We also show that a linear map from a Euclidean space into Sol or Nil space is biharmonic if and only if it is a harmonic map, and give a complete classification of such maps.
Biharmonic maps are the critical points of the bienergy functional and generalise harmonic maps. We investigate the index of a class of biharmonic maps, derived from minimal Riemannian immersions into spheres. This study is motivated by three families of examples: the totally geodesic inclusion of spheres, the Veronese map and the Clifford torus.
On p-biharmonic curves
本文研究$p$-双调和曲线，推广双调和曲线，分类闭曲面与三维空间形式上的$\ rac{1}{2}$-双调和曲线，证明其存在性并讨论稳定性，揭示与$p$-弹性曲线的联系。
Homogeneity of magnetic geodesics in the Heisenberg group
研究Heisenberg群中由标准接触结构导出的磁流形的齐性性质，证明其具有齐性特征。
We consider Heisenberg groups equipped with a sub-Finsler metric. Using methods of optimal control theory we prove that in this geometric setting the infinite geodesics are horizontal lines under the assumption that the sub-Finsler metric is defined by a strictly convex norm. This answers a question posed in [5] and has applications in the characterisation of isometric embeddings into Heisenberg groups.
In this note we prove that the Heisenberg group with a left-invariant pseudo-Riemannian metric admits a completely integrable totally geodesic distribution of codimension 1. This is on the contrary to the Riemannian case, as it was proved by T. Hangan.
In this paper we study the notion of geodesic curvature of smooth horizontal curves parametrized by arc lenght in the Heisenberg group, that is the simplest sub-Riemannian structure. Our goal is to give a metric interpretation of this notion of geodesic curvature as the first corrective term in the Taylor expansion of the distance between two close points of the curve.
A characterization of horizontally totally geodesic hypersurfaces in Heisenberg groups
针对高维Heisenberg群的Bernstein问题，本文通过刻画直纹性质并研究次Riemannian测地线，证明ℍⁿ（n≥2）中水平对称第二基本形式为零的整体超曲面只能是超平面。
We prove that the horizontal and vertical distributions of the tangent bundle with the Sasaki metric are isocline, the distributions given by the kernels of the horizontal and vertical lifts of the contact form $ω$ from the Heisenberg manifold $(H_3,g)$ to $(TH_3,g^S)$ are not totally geodesic, and the distributions $F^H=L(E_1^H,E_2^H)$ and $F^V=L(E_1^V,E_2^V)$ are totally geodesic, but they are not isocline. We obtain that the horizontal and natural lifts of the curves from the Heisenberg manifold $(H_3,g)$, are geodesics in the tangent bundle endowed with the Sasaki metric $(TH_3,g^s)$, if a…
Classification of the geodesic curves of the sub-Riemannian LR system on the Heisenberg group in dimension 5
本文研究了五维Heisenberg群上子Riemannian LR系统测地曲线的分类及完全可积哈密顿系统的解法。
Sub-Riemannian LR geodesic flows on Heisenberg groups
本文研究海森堡群上的次黎曼LR测地流，区分其低维与高维可积性，将结果与LL系统比较以明确两类测地流的异同。
Closed Magnetic geodesics on Heisenberg nilmanifolds
本文研究了在三维Heisenberg nilmanifolds上对于每一个左不变洛伦兹力存在闭磁测地线的问题，主要贡献是证明了在紧凑商 manifold M 上对于任意能量水平低于Mañé临界值总存在可缩的闭磁测地线，同时展示了某些例子中不存在非可缩的闭磁测地线。
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.
This paper is part of an undergraduate research project. We discuss the Heisenberg group H1, the three-dimensional space R3 equipped with one of two equivalent metrics, the Koranyi- and Carnot- Caratheodory metric. We show that the notion of length of curves for both metrics coincide, and that shortest curves, so-called geodesics, exist.
We study the dynamics of magnetic flows on Heisenberg groups. Let $H$ denote the three-dimensional simply connected Heisenberg Lie group endowed with a left-invariant Riemannian metric and an exact, left-invariant magnetic field. Let $Γ$ be a lattice subgroup of $H,$ so that $Γ\backslash H$ is a closed nilmanifold. We first find an explicit description of magnetic geodesics on $H$, then determine all closed magnetic geodesics and their lengths for $Γ\backslash H$. We then consider two applications of these results: the density of periodic magnetic geodesics and marked magnetic length spectrum …
In this paper, on the first, we prove $Δr=2H$ where $Δ$ is the Laplacian operator, $r=\left( r_{1},r_{2},r_{3}\right) $ the position vector field and $H$ is the mean curvature vector field of a surface $\mathcal{S}$ in the 3-dimensional Heisenberg group $H_{3}.$ In the second, we classify the ruled surfaces by straight geodesic lines, which are of finite type in $H_{3}.$ The straight geodesic lines belong to $\ker ω,$ where $ω$ is the Darboux form.
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.
Sub-Riemannian geodesics on the Heisenberg 3D nil-manifold
本文研究了3D Heisenberg群上次黎曼结构到Heisenberg 3D nil-manifold的投影，分析了测地流的动力学性质，并基于$G$中次黎曼球与距离的精确双边界估计了$M$中测地线的切割时间。
Clairaut anti-invariant Riemannian maps to Sasakian manifolds
本文研究了底空间为Sasakian流形的Clairaut反不变Riemannian映射的几何，给出了曲线为测地线及映射为Clairaut的条件，并探讨了双调和性与示例。
Measure contraction property, curvature exponent and geodesic dimension of sub-Finsler $\ell^p$-Heisenberg groups
该研究首次探索次芬斯勒几何的合成曲率-维数界，分析ℓ^p-海森堡群的测度收缩性质与测地线维数，明确不同p取值下的性质成立条件，并给出首个曲率指数与测地线维数存在间隙的度量测度空间实例。
It is shown that parts of planes, helicoids and hyperbolic paraboloids are the only minimal surfaces ruled by geodesics in the three dimensional Riemannian Heisenberg group. It is also shown that they are the only surfaces in the three dimensional Heisenberg group whose mean curvature is zero with respect to both of the standard Riemannian metric and the standard Lorentzian metric.
The main focus of the paper is the investigation of moduli space of left invariant pseudoRiemannian metrics on the cotangent bundle of Heisenberg group. Consideration of orbits of the automorphism group naturally acting on the space of the left invariant metrics allows us to use the algebraic approach. However, the geometrical tools, such as classification of hyperbolic plane conics, will often be required. For metrics that we obtain in the classification, we investigate geometrical properties: curvature, Ricci tensor, sectional curvature, holonomy and parallel vector fields. The classificatio…
We use a Riemannnian approximation scheme to define a notion of $\textit{sub-Riemannian Gaussian curvature}$ for a Euclidean $C^{2}$-smooth surface in the Heisenberg group $\mathbb{H}$ away from characteristic points, and a notion of $\textit{sub-Riemannian signed geodesic curvature}$ for Euclidean $C^{2}$-smooth curves on surfaces. These results are then used to prove a Heisenberg version of the Gauss-Bonnet theorem. An application to Steiner's formula for the Carnot-Carathéodory distance in $\mathbb{H}$ is provided.
In this paper, we compute sub-Riemannian limits of Gaussian curvature associated to two kinds of Schouten-Van Kampen affine connections and the adapted connection for a Euclidean $C^2$-smooth surface in the Heisenberg group away from characteristic points and signed geodesic curvature associated to two kinds of Schouten-Van Kampen affine connections and the adapted connection for Euclidean $C^2$-smooth curves on surfaces. We get Gauss-Bonnet theorems associated to two kinds of Schouten-Van Kampen affine connections in the Heisenberg group.
On the sub-Riemannian geometry of the quaternionic Heisenberg group
本文研究四元数Heisenberg群的子黎曼几何，通过定义一系列黎曼度量并分析其极限，刻画了测地线和距离，并推导了超曲面的水平平均曲率公式。
Geodesic orbit pseudo-Riemannian H-type nilmanifolds: case of minimal admissible Clifford modules
本文研究了伪黎曼H型nilmanifold的测地轨道性质，扩展了Riehm关于黎曼H型李群的研究，并针对由最小维度可允许克利福德模块构造的李代数提供了完整特征化。
Magnetic equations on the Heisenberg group: symmetries, solutions and the inverse problem of the calculus of variations
本文研究Heisenberg群上磁方程的对称性、解及变分学逆问题，通过显式计算磁轨迹并展示其作为变分问题解的性质。
For the ``Hopf bundle'' $S^1\to S^{2n,1} \to {\mathbb H}^n$, horizontal lifts of simple closed curves are studied. Let $γ$ be a piecewise smooth, simple closed curve on a complete totally geodesic surface $S$ in the base space. Then the holonomy displacement along $γ$ is given by $$ V(γ)=e^{λA(γ) i} $$ where $A(γ)$ is the area of the region on the surface $S$ surrounded by $γ$; $λ=1/2 $ or 0 depending on whether $S$ is a complex submanifold or not. We also carry out a similar investigation for the complex Heisenberg group ${\mathbb R} \to {\mathcal H}^{2n+1} \to {\mathbb C}^n$.
Computational geometry in Heisenberg group Heis^3
本文研究海森堡群 Heis^3 中的计算几何，利用左不变度量和联络曲率结果求解测地线方程，并用 Mathematica 绘制测地线与度量球。
We prove that no Brunn--Minkowski inequality from the Riemannian theories of curvature-dimension and optimal transportation can by satisfied by a strictly subRiemannian structure. Our proof relies on the same method as for the Heisenberg group together with new investigations by Agrachev, Barillari and Rizzi on ample normal geodesics of subRieman-nian structures and the geodesic dimension attached to them.
Local control on quaternionic Heisenberg group of dimension $7$
该研究将7维四元数Heisenberg群刻画为矩阵群，探究其左不变控制系统的局部控制特性，分析次黎曼结构对称性对测地线最优性的作用。
We characterize convex isoperimetric sets in the Heisenberg group endowed with horizontal perimeter. We first prove Sobolev regularity for a certain class of vector fields in the plane with bounded variation, related to the curvature equations. Then, by an approximation-reparameterization argument, we show that the boundary of convex isoperimetric sets is foliated by geodesics of the Carnot-Caratheodory distance.
We study the geodesics problem in Heisenberg group H (case SR and riemannian). The sheaf of infinitesimal automorphisms of the (2n,2n+1) distribution D over H is an infinite, transitive Lie algebra sheaf.
The generalised energy identity and length of necks for $\varepsilon$-harmonic maps
本文研究了ε-调和映射的广义能量恒等式和颈部存在性，通过特定量确定能量恒等式和颈部形成条件，并在失效时计算能量损失和颈部长度。
It is proved that the Heisenberg group $\operatorname*{Nil}\nolimits_{3}$ with a balanced metric, the sum of the left and right invariant metrics, splits as a Riemannian product $\mathbb{T\times Z}$, where $\mathbb{T}$ is a totally geodesic surface and $\mathbb{Z}$ the center of $\operatorname*{Nil}% \nolimits_{3}.$ It is then proved the existence of complete properly embedded minimal surfaces in $\operatorname*{Nil}\nolimits_{3}$ by solving the asymptotic Dirichlet problem for the minimal surface equation on $\mathbb{T}$. It is also proved the existence of complete properly embedded minimal s…
On $f$-polyharmonic maps between Riemannian manifolds
研究黎曼流形之间的$f$-多重调和映射，推导其欧拉-拉格朗日方程，对特定空间形式中具有正常测地曲率的$f$-$k$-调和曲线分类，给出$f$-多重调和函数和映射构造并证明刘维尔型定理。
Given a negatively curved geodesic metric space M, we study the asymptotic penetration behaviour of geodesic lines of M in small neighbourhoods of closed geodesics and of other compact convex subsets of M. We define a spiraling spectrum which gives precise information on the asymptotic spiraling lengths of geodesic lines around these objects. We prove analogs of the theorems of Dirichlet, Hall and Cusick in this context. As a consequence, we obtain Diophantine approximation results of real numbers, complex numbers, or elements of the Heisenberg group by irrational quadratic ones.
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。
Normal geodesic flows flows of Carnot-Caratheodory are discussed from the point of view of the theory of Hamiltonian systems. The geodesic flows corresponding to left-invariant metrics and left- and -right-invariant rank 2 distributions on the three-dimensional Heisenberg group are analysed as integrable systems. The flows corresponding to left-invariant metrics and left-invariant distributions on Lie groups are reduced to Euler equations on Lie groups. Relation of these constructions to problems of analytical mechanics is discussed.
We classify all self-dual Einstein four-manifolds invariant under a principal action of the three-dimensional Heisenberg group with non-degenerate orbits. The metrics are explicit and we find, in particular, that the Einstein constant can take any value. Then we study when the corresponding (Riemannian or neutral-signature) metrics are (geodesically) complete. Finally, we exhibit the solutions of non-zero Ricci-curvature as different branches of one-loop deformed universal hypermultiplets in Riemannian and neutral signature.
In this note, we show that sub-Riemannian manifolds can contain branching normal minimizing geodesics. This phenomenon occurs if and only if a normal geodesic has a discontinuity in its rank at a non-zero time, which in particular for a strictly normal geodesic means that it contains a non-trivial abnormal subsegment. The simplest example is obtained by gluing the three-dimensional Martinet flat structure with the Heisenberg group in a suitable way. We then use this example to construct more general types of branching.
In this paper we study the geodesic flow on nilmanifolds equipped with a left-invariant metric. We write the underlying definitions and find general formulas for the Poisson involution. As an example we develop the Heisenberg Lie group equipped with its canonical metric. We prove that a family of first integrals giving the complete integrability can be read off at the Lie algebra of the isometry group. We also explain the complete integrability on compact quotients and for any invariant metric.
By adapting a technique of Molchanov, we obtain the heat kernel asymptotics at the sub-Riemannian cut locus, when the cut points are reached by an $r$-dimensional parametric family of optimal geodesics. We apply these results to the bi-Heisenberg group, that is, a nilpotent left-invariant sub-Rieman\-nian structure on $\mathbb{R}^{5}$ depending on two real parameters $α_{1}$ and $α_{2}$. We develop some results about its geodesics and heat kernel associated to its sub-Laplacian and we illuminate some interesting geometric and analytic features appearing when one compares the isotropic ($α_{1}=…
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.
In the infinite dimensional Heisenberg group, we construct a left invariant weak Riemannian metric that gives a degenerate geodesic distance. The same construction yields a degenerate sub-Riemannian distance. We show how the standard notion of sectional curvature adapts to our framework, but it cannot be defined everywhere and it is unbounded on suitable sequences of planes. The vanishing of the distance precisely occurs along this sequence of planes, so that the degenerate Riemannian distance appears in connection with an unbounded sectional curvature. In the 2005 paper by Michor and Mumford,…
The present paper starts with an introduction to quaternions and then defines the 3-dimmensional sphere as the set of quaternions of length one. The quaternion group induces on $\mathbb{S}^3$ a structure of noncommutative Lie group. This group is compact and the results obtained in this case are very different than those obtained in the case of the Heisenberg group, which is a noncompact Lie group. Like in the Heisenberg group case, we introduce a nonintegrable distribution on the sphere and a metric on it using two of the noncommutative left invariant vector fields. This way $\mathbb{S}^3$ be…
This work is an investigation of perimeter measures in the metric measure space given by the Heisenberg group with Haar measure and a Carnot-Carathéodory metric, which is in general a sub-Finsler metric. Included is a reduction of Minkowski content in any CC-metric to an integral formula in terms of Lebesgue surface area in $\mathbb{R}^3$. Using this result, I study two perimeter measures that arise from the study of Finsler normed planes, and provide evidence that Pansu's conjecture regarding the isoperimetric problem in the sub-Riemannian case appears to hold in the more general sub-Finsler …
Sub-Riemannian geodesics in the affine-additive group
本文利用庞特里亚金最大值原理推导仿射加法群的次黎曼测地线，计算割时间，并构造Carnot-Carathéodory球面及烧瓶曲面，后者为海森堡群中Pansu球面的类似物。
We introduce a notion of geodesic curvature $k_ζ$ for a smooth horizontal curve $ζ$ in a three-dimensional contact sub-Riemannian manifold, measuring how much a horizontal curve is far from being a geodesic. We show that the geodesic curvature appears as the first corrective term in the Taylor expansion of the sub-Riemannian distance between two points on a unit speed horizontal curve $$ d_{SR}^2( ζ(t),ζ(t+ε))=ε^2-\frac{k_ζ^2(t)}{720} ε^6 +o(ε^{6}). $$ The sub-Riemannian distance is not smooth on the diagonal, hence the result contains the existence of such an asymptotics. This can be seen as …
We begin a systematic study of these spaces, initially following along the lines of Eberlein's comprehensive study of the Riemannian case. In particular, we integrate the geodesic equation, discuss the structure of the isometry group, and make a study of lattices and periodic geodesics. Some major differences from the Riemannian theory appear. There are many flat groups (versus none), including Heisenberg groups. While still a semidirect product, the isometry group can be strictly larger than the obvious analogue. Everything is illustrated with explicit examples. We introduce the notion of pH-…
We initiate the study of X-ray tomography on sub-Riemannian manifolds, for which the Heisenberg group exhibits the simplest nontrivial example. With the language of the group Fourier Transform, we prove an operator-valued incarnation of the Fourier Slice Theorem, and apply this new tool to show that a sufficiently regular function on the Heisenberg group is determined by its line integrals over sub-Riemannian geodesics. We also consider the family of taming metrics $g_ε$ approximating the sub-Riemannian metric, and show that the associated X-ray transform is injective for all $ε>0$. This resul…
Magnetic Brunn-Minkowski inequalities
研究黎曼流形上由磁性测地线定义的闵可夫斯基平均，建立Brunn-Minkowski不等式与磁性里奇曲率下界的等价性，并在海森堡群上证明尖锐的接触磁性Brunn-Minkowski不等式。
Sub-Riemannian and sub-Lorentzian geodesics of the oscillator groups
本文研究振子群的次黎曼与次洛伦兹测地线，推广至半单李群与紧子群对应的Osc(G,K)，明确其测地线公式并实现为斯蒂弗尔丛。
Harmonic maps into sub-Riemannian Lie groups
该研究推广黎曼流形的相关定义，提出次黎曼流形间调和映射的定义，推导了映到左不变度量李群的水平映射为调和映射的条件，证明其可分为异常与正常两类，并以Heisenberg群的调和映射方程为例进行说明。
A characterization of maximal domains of existence of adapted complex structures for Riemannian homogeneous manifolds under certain extensibility assumptions on their geodesic flow is given. This is applied to generalized Heisenberg groups and naturally reductive Riemannian homogeneous spaces. As an application it is shown that the case of generalized Heisenberg groups yields examples of maximal domains of definition for the adapted complex structure which are are neither holomorphically separable, nor holomorphically convex.
We study left-invariant distances on Lie groups for which there exists a one-parameter family of homothetic automorphisms. The main examples are Carnot groups, in particular the Heisenberg group with the standard dilations. We are interested in criteria implying that, locally and away from the diagonal, the distance is Euclidean Lipschitz and, consequently, that the metric spheres are boundaries of Lipschitz domains in the Euclidean sense. In the first part of the paper, we consider geodesic distances. In this case, we actually prove the regularity of the distance in the more general context o…
We compute the full holonomy group of compact Lorentzian manifolds with parallel Weyl tensor, which are neither conformally flat nor locally symmetric, for the case where the fundamental group is contained in a distinguished subgroup G of the isometry group of the universal cover. To prove this, we show that every such compact Lorentzian manifold has to be geodesically complete. Moreover we characterize the identity component of the isometry group for this universal cover and show that G is up to a discrete factor contained in the latter. Concretely, we prove that the identity component of the…
扫码添加微信好友，提出您的宝贵建议 👇
💡 备注请填写：网站反馈
发表机构由 AI 从论文文本抽取，如发现错误，欢迎反馈修正建议。