Geodesic Flow on the Normal Congruence of a Minimal Surface

仿射距离函数与平面汇的通用奇点

Generic singularities of affine distance functions and plane congruences

本文分类了$\mathbb{R^4}$中2参数平面汇和仿射法平面汇的通用奇点，并研究了仿射距离函数族的通用奇点。

Surfaces with flat normal bundle: an explicit construction

An explicit construction of surfaces with flat normal bundle in the Euclidean space (unit hypersphere) in terms of solutions of certain linear system is proposed. In the case of 3-space our formulae can be viewed as the direct Lie sphere analog of the generalized Weierstrass representation of surfaces in conformal geometry or the Lelieuvre representation of surfaces in the affine space. An explicit parametrization of Ribaucour congruences of spheres by three solutions of the linear system is obtained. In view of the classical Lie correspondence between Ribaucour congruences and surfaces with f…

$\u211d^4$中三参数线汇的奇点

Singularities of 3-parameter line congruences in $\mathbb{R}^4$

该研究针对$\u211d^4$中的三参数线汇奇点问题，完成了这类线汇奇点及Blaschke仿射法线汇奇点的通有分类工作。

Line Congruences on singular surfaces

This paper is a first step in order to extend Kummer's theory for line congruences to the case $\lbrace x, ξ\rbrace $, where $x: U \rightarrow \mathbb{R}^3$ is a smooth map and $ξ: U \rightarrow \mathbb{R}^3$ is a proper frontal. We show that if $\lbrace x, ξ\rbrace$ is a normal congruence, the equation of the principal surfaces is a multiple of the equation of the developable surfaces, furthermore, the multiplicative factor is associated to the singular set of $ξ$.

Pontryagin maximum principle, (co)adjoint representation, and normal geodesics of left-invariant (sub-)Finsler metrics on Lie groups

On the ground of origins of the theory of Lie groups and Lie algebras, their (co)adjoint representations, and the Pontryagin maximum principle for the time-optimal problem are given an independent foundation for methods of geodesic vector field to search for normal geodesics of left-invariant (sub-)Finsler metrics on Lie groups and to look for the corresponding locally optimal controls in (sub-)Riemannian case, as well as some their applications.

Normal geodesics connecting two non-necessarily spacelike submanifolds in a stationary spacetime

In this paper we obtain an existence theorem for normal geodesics joining two given submanifolds in a globally hyperbolic stationary spacetime. The proof is based on both variational and geometric arguments involving the causal structure of the spacetime, the completeness of suitable Finsler metrics associated to it and some basic properties of a submersion. By this interaction, unlike previous results on the topic, also non--spacelike submanifolds can be handled.

Pseudo-Riemannian Submanifolds with 3-Planar Geodesics

In the present paper we study pseudo-Riemannian submanifolds which have 3-planar geodesic normal sections.We consider W-curves (helices) on pseudo-Riemannian submanifolds. Finally, we give neccessary and sufficient condition for a normal section to be a W-curve on pseudo-Riemannian submanifolds.

伪黎曼子流形的测地法坐标与自然张量

Geodesic normal coordinates and natural tensors for pseudo-Riemannian submanifolds

本文构造了适用于伪黎曼子流形的测地法坐标，证明度量泰勒系数可由背景曲率张量与第二基本形式的协变导数的通用多项式表示，并据此定义和刻画了自然子流形张量与微分算子。

A note on Carnot geodesics in nilpotent Lie groups

We show that strictly abnormal geodesics arise in graded nilpotent Lie groups. We construct such a group, for which some Carnot geodesics are strictly abnormal; in fact, they are not normal in any subgroup. In the step-2 case we also prove that these geodesics are always smooth. Our main technique is based on the equations for the normal and abnormal curves, that we derive (for any Lie group) explicitly in terms of the structure constants.

Logarithmic growth of systole of arithmetic Riemann surfaces along congruence subgroups

We apply a study of orders in quaternion algebras, to the differential geometry of Riemann surfaces. The least length of a closed geodesic on a hyperbolic surface is called its systole, and denoted syspi_1. P. Buser and P. Sarnak constructed Riemann surfaces X whose systole behaves logarithmically in the genus g(X). The Fuchsian groups in their examples are principal congruence subgroups of a fixed arithmetic group with rational trace field. We generalize their construction to principal congruence subgroups of arbitrary arithmetic surfaces. The key tool is a new trace estimate valid for an arb…

多元正态分布的测地线与Toda晶格型Lax对

Geodesics of multivariate normal distributions and a Toda lattice type Lax pair

本文研究多元正态分布在Fisher度量下的测地线，利用块Cholesky分解和黎曼淹没给出计算公式与中点算法，并由此导出新的Toda晶格型Lax对。

No Periodic normal Geodesics in $J^k(\mathbb{R},\mathbb{R}^n)$

The space of $k$-jets of $n$ real function of one real variable $x$ admits the structure of a Carnot group, which then has an associated Hamiltonian geodesic flow. As in any Hamiltonian flow, a natural question is the existence of periodic solutions. Does the space of $k$-jets have periodic geodesics? This study will demonstrate the integrability of subRiemannian geodesic flow, characterize and classify the subRiemannian geodesics in the space of $k$-jets, and show that they are never periodic.

共形通用子黎曼度量的正则奇异测地线

Normal Singular Geodesics of a Conformally Generic Sub-Riemannian Metric

该研究证明了在特定分布下，即使加入势函数，也不存在非平凡的正则奇异轨道。

Branching geodesics in sub-Riemannian geometry

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.

Conformal Rigidity from Focusing

The null curvature condition (NCC) is the requirement that the Ricci curvature of a Lorentzian manifold be nonnegative along null directions, which ensures the focusing of null geodesic congruences. In this note, we show that the NCC together with the causal structure significantly constrain the metric. In particular, we prove that any conformal rescaling of a vacuum spacetime introduces either geodesic incompleteness or negative null curvature, provided the conformal factor is non-constant on at least one complete null geodesic. In the context of bulk reconstruction in AdS/CFT, our results co…

Local normal forms for geodesically equivalent pseudo-Riemannian metrics

Two pseudo-Riemannian metrics $g $ and $\bar g$ are geodesically equivalent, if they share the same (unparameterized) geodesics. We give a complete local description of such metrics which solves the natural generalisation of Beltrami problem for pseudo-Riemannian metrics.