David Bao, Colleen Robles, Zhongmin Shen
In this paper, we study Zermelo navigation on Riemannian manifolds and use that to solve a long standing problem in Finsler geometry. Namely, the complete classification of strongly convex Randers metrics of constant flag curvature.
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The present paper studies globally defined Kropina metrics as solutions of the Zermelo's navigation problem. Moreover, we characterize the Kropina metrics of constant flag curvature showing that up to local isometry, there are only two model spaces of them: the Euclidean space and the odd-dimensional spheres.
Capillary hypersurfaces, Heintze-Karcher's inequality and Zermelo's navigation
本文通过Zermelo导航构造特殊Finsler度量，研究其测地法向流，从而建立单位球中毛细超曲面的Heintze-Karcher型不等式，揭示毛细超曲面与自由边界超曲面之间的对应关系。
The generalized Zermelo navigation problem looks for the shortest time paths in an environment, modeled by a Finsler manifold (M,F), under the influence of wind or current, represented by a vector field W. The main objective of this paper is to investigate the relationship between the isoparametric functions on the manifold M with and without the presence of the vector field W. For the positive-definite cases, we also compare the mean curvatures in the manifold. Overall, we follow a coordinate-free approach.
We generalize and study the Zermelo navigation problem on Hermitian manifolds in the presence of a perturbation $W$ determined by a mild complex velocity vector field $||W(z)||_h<||u(z)||_h$, with application of complex Finsler metric of complex Randers type. By admitting space-dependence of ship's relative speed $||u(z)||_h\leq1$ we discuss the projectively related complex Finsler metrics, the geodesics corresponding to the solutions of Zermelo's problem, in particular the conformal case and the connections between the corresponding background Hermitian metric $h$, new Hermitian metric $a$ an…
We generalize the notion of Zermelo navigation to arbitrary pseudo-Finsler metrics possibly defined in conic subsets. The translation of a pseudo-Finsler metric $F$ is a new pseudo-Finsler metric whose indicatrix is the translation of the indicatrix of $F$ by a vector field $W$ at each point, where $W$ is an arbitrary vector field. Then we show that the Matsumoto tensor of a pseudo-Finsler metric is equal to zero if and only if it is the translation of a semi-Riemannian metric, and when $W$ is homothetic, the flag curvature of the translation coincides with the one of the original one up to th…
In this article, based on two case studies, we discuss the role of abnormal geodesics in planar Zermelo navigation problems. Such curves are limit curves of the accessibility set, in the domain where the current is strong. The problem is set in the frame of geometric time optimal control, where the control is the heading angle of the ship and in this context, abnormal curves are shown to separate time minimal curves from time maximal curves and are both small-time minimizing and maximizing. We describe the small-time minimal balls. For bigger time, a cusp singularity can occur in the abnormal …
We generalize the Zermelo navigation problem and its solution on Riemannian manifolds $(M, h)$ admitting a space dependence of a ship's own speed $|u(x)|_h\leq1$ in the presence of a perturbation $W$ determined by a mild velocity vector field $|W(x)|_h<|u(x)|_h$, with application of Finsler metric of Randers type.
We generalize the Zermelo navigation problem and its solution on Riemannian manifolds $(M, h)$ admitting a space dependence of a ship's speed $0<|u(x)|_h\leq1$ in the presence of a perturbation $\tilde{W}$ determined by a strong velocity vector field satisfying $|\tilde{W}(x)|_h=|u(x)|_h$, with application of Finsler metric of Kropina type.
The goal of this paper is to describe Zermelo's navigation problem on Riemannian manifolds as a time-optimal control problem and give an efficient method in order to evaluate its control curvature. We will show that up to change the Riemannian metric on the manifold the control curvature of Zermelo's problem has a simple to handle expression which naturally leads to a generalization of the classical Gauss-Bonnet formula in an inequality. This Gauss-Bonnet inequality enables to generalize for Zermelo's problems the E. Hopf theorem on flatness of Riemannian tori without conjugate points.
Time-dependent Zermelo navigation with tacking
在 Lorentz-Finsler 几何框架下研究时间与位置依赖的 Zermelo 导航问题，推导了仅时间依赖情形的全局结果及非凸导航下的换舷行为，并提出高效算法求解一般非凸设定下的时间最小化轨迹边值问题。
On the geometry of Zermelo's optimal control trajectories
本文从几何与微分方程视角研究 Zermelo 导航问题，导出最优轨迹满足的精确 ODE 系统，并利用外部场的一阶近似实现数值求解，从而更准确高效地优化代价函数。
Wind Finslerian structures: from Zermelo's navigation to the causality of spacetimes
本文提出风 Finsler 结构，统一了 Zermelo 导航问题与具有非零 Killing 向量场的时空因果结构描述，并给出导航解、度量性质及一般 Fermat 原理等应用。
In certain circumstances tools of Riemannian geometry are sufficient to address questions arising in the more general Finslerian context. We show that one such instance presents itself in the characterisation of geodesics in Randers spaces of constant flag curvature. To achieve a simple, Riemannian derivation of this special family of curves, we exploit the connection between Randers spaces and the Zermelo problem of time-optimal navigation in the presence of background fields. The characterisation of geodesics is then proven by generalising an intuitive argument developed recently for the sol…
We consider a triality between the Zermelo navigation problem, the geodesic flow on a Finslerian geometry of Randers type, and spacetimes in one dimension higher admitting a timelike conformal Killing vector field. From the latter viewpoint, the data of the Zermelo problem are encoded in a (conformally) Painleve-Gullstrand form of the spacetime metric, whereas the data of the Randers problem are encoded in a stationary generalisation of the usual optical metric. We discuss how the spacetime viewpoint gives a simple and physical perspective on various issues, including how Finsler geometries wi…
Time geodesics on a slippery cross slope under gravitational wind
本文研究任意维湿滑山坡在横向重力风下的时间最优导航，用新型一般 (α, β) Finsler 度量给出纯几何解，并通过二维示例分析牵引与风力影响。
Superwind and navigation of least time on Riemannian manifolds
利用芬斯勒几何分析黎曼流形上的最短时间导航，引入超风概念涵盖多种情况，推广现有理论，给出导航问题解及强凸性条件，还探索二维示例。
A general model for time-minimizing navigation on a mountain slope under gravity
本文以 Riemann-Finsler 几何推广 Matsumoto 山坡问题，通过度量各向异性变形和重力风平移构造一般 $(α, β)$-度量，给出重力下时间最短导航的几何解与二维验证。
We consider the Zermelo navigation problem on the ellipsoid of revolution (spheroid) in the presence of a perturbation $W$ determined by a mild velocity vector field, $|W|<1$, with application of Finsler metric of Randers type in the context of the corresponding optimal control represented by a time-efficient ship's heading $φ(t)$ (steering direction). As the example we present the solutions to the problem on an oblate ellipsoid with acting infinitesimal rotation.
An account on links between Finsler and Lorentz Geometries for Riemannian Geometers
本文面向 Riemannian 几何学家介绍 Lorentz 与 Finsler 几何间的联系，以 Zermelo 导航问题为例，阐述了利用 Finsler 元素描述 Lorentz 因果关系、处理 Finsler 奇点及因果边界，并应用于波与火线传播。
Multiple connecting geodesics of a Randers-Kropina metric via homotopy theory for solutions of an affine control system
本文借助 Lusternik-Schnirelman 理论及受控漂移仿射控制系统解空间的同伦结果，证明不可缩流形上任意两点间存在无穷多条 Randers-Kropina 测地线。
Geometric and topological properties of manifolds in robot motion planning
本综述探讨了机器人运动规划中流形的几何与拓扑性质，涵盖拓扑复杂度的计算与估计、黎曼流形中的测地运动规划，以及临界点理论与流形拓扑的联系，并分析了导航函数与球面复杂度的关系。
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