Jeanne N. Clelland, Christopher G. Moseley
We define the notion of sub-Finsler geometry as a natural generalization of sub-Riemannian geometry with applications to optimal control theory. We compute a complete set of local invariants, geodesic equations, and the Jacobi operator for the three-dimensional case and investigate homogeneous examples.
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
A geometric approach to differential game theory is illustrated. The parallel pursuit is considered as a two-player zero-sum differential game. The optimal strategies of each player is designed based on Riemann-Finsler geometry. Our approach incorporates a closed loop optimal control and the presentation is familiar with a Brachistochrone type problem.
A new convenient method of describing flat convex compact sets is proposed. It generalizes classical trigonometric functions $\sin$ and $\cos$. Apparently, this method may be very useful for explicit description of solutions of optimal control problems with two-dimensional control. Using this method a series of sub-Finsler problems with two-dimensional control lying in an arbitrary convex set $Ω$ is investigated. Namely, problems on the Heisenberg, Engel, and Cartan groups and also Grushin's and Martinet's cases are considered. A particular attention is paid to the case when $Ω$ is a polygon.
The geometry and analysis on Finsler manifolds is a very important part of Finsler geometry. In this article, we introduce some important and fundamental topics in global Finsler geometry and discuss the related properties and the relationships in them. In particular, we optimize and improve the various definitions of Lie derivatives on Finsler manifolds. We also characterize the gradient vector fields and obtain a gradient estimate for any smooth function on a Randers manifold.
As a natural application of the {\it theory of geometric averaging} in Finsler geometry and generalized Finsler geometry, a new approach to investigate {\it generalized Finsler geometry}, based on a convex invariance of the average structures, is introduced.
Here, a non-linear analysis method is applied rather than classical one to study projective Finsler geometry. More intuitively, by means of an inequality on Ricci-Finsler curvature, a projectively invariant pseudo-distance is introduced and an analogous of Schwarz' lemma in Finsler geometry is proved. Next, the Schwarz' lemma is applied to show that the introduced pseudo-distance is a distance. This projectively invariant distance will be served in continuation of this work to investigate Einstein-Finsler spaces and classify Finsler spaces as well.
Here, an axiom of spheres in Finsler geometry is proposed and it is proved that if a Finslerian manifold satisfies the axiom of spheres then it is of constant flag curvature.
Adopting the pullback formalism, a new linear connection in Finsler geometry has been introduced and investigated. Such connection unifies all formerly known Finsler connections and some other connections not introduced so far. Also, our connection is a Finslerian version of the Tripathi connection introduced in Riemannian geometry. The existence and uniqueness of such connection is proved intrinsically. An explicit intrinsic expression relating this connection to Cartan connection is obtained. Some generalized Finsler connections are constructed from Tripathi Finsler connection, by applying t…
Lectures on Finslerian Geometry
本讲义系统介绍Finsler几何的基本概念与理论框架，为学习者提供该领域的核心知识。
A method to generalize results from Riemannian Geometry to Finsler geometry is presented. We use the method to generalize several results that involve only metric conditions. Between them we show that the topology induced by the Finsler structure is equivalent to the manifold topology, we provide a new proof of the Hopf-Rinow theorem in Finsler geometry and we prove the existence of the center of mass of a convex body when the non-symmetric distance function comes from a non-reversible Finsler function.
We prove Hessian comparison theorems, Laplacian comparison theorems and volume comparison theorems of Finsler manifolds under various curvature conditions. As applications, we derive Mckean type theorems for the first eigenvalue of Finsler manifolds, as well as generalize a result on fundamental group due to Milnor to Finsler manifolds.
In this short paper, we study a symmetric covariant tensor in Finsler geometry, which is called the mean Berwald curvature. We first investigate the geometry of the fibres as the submanifolds of the tangent sphere bundle on a Finsler manifold. Then we prove that if the mean Berwald curvature is isotropic along fibres, then the Berwald scalar curvature is constant along fibres.
In this paper, we first establish an equivalence theorem of Minkowski spaces by using results in centro-affine differential geometry. As an application in Finsler geometry, we gives some new characterizations of Berwald spaces.
In this paper, we generalize the classification of geodesic orbit spheres from Riemannian geometry to Finsler geometry. Then we further prove if a geodesic orbit Finsler sphere has constant flag curvature, it must be Randers. It provides an alternative proof for the classification of invariant Finsler metrics with $K\equiv1$ on homogeneous spheres other than $Sp(n)/Sp(n-1)$.
The sub-Finslerian geometry means that the metric $F$ is defined only on a given subbundle of the tangent bundle, called a horizontal bundle. In the paper, a version of the Hopf-Rinow theorem is proved in the case of sub-Finslerian manifolds, which relates the properties of completeness, geodesically completeness, and compactness. The sub-Finsler bundle, the exponential map and the Legendre transformation are deeply involved in this investigation.
I show that Matsumoto conjectured inequality between relative length and Finsler length is false. The incorrectness of the claim is easily inferred from the geometry of the indicatrix.
In this short note, we prove that if $F$ is a weak upper semicontinuous admissible Finsler structure on a domain in $\mathbb{R}^n$, $n\geq 2$, then the intrinsic distance and differential structures coincide.
In this paper, we establish two Santaló type formulas for general Finsler manifolds. As applications, we derive a universal lower bound for the first eigenvalue of the nonlinear Laplacian, two Croke type isoperimetric inequalities, and a Yamaguch type finiteness theorem in Finser geometry.
Based on a self-contained, coordinate-free exposition of the necessary concepts and tools of spray and Finsler geometry (with detailed proofs), we derive new results among others on the consequences of the direction-independence of the Landsberg tensor and the stretch tensor of a Finsler manifold. We show that an at least 3-dimensional Finsler manifold with vanishing projected Berwald tensor is a Berwald manifold. To obtain consequences in two dimensions, we transcript Berwald's classical theory on 2-dimensional Finsler manifolds in our setup. We prove criteria and necessary conditions for Fin…
We propose a unified approach to the theory of connections in the geometry of sprays and Finsler metrics which, in particular, gives a simple explanation of the well-known fact that all the classical Finslerian connections provide exactly the same formulas appearing in the calculus of variations.
In complex Finsler geometry, an open problem is: does there exist a weakly Kähler Finsler metric which is not Kähler? In this paper, we give an affirmative answer to this open problem. More precisely, we construct a family of the weakly Kähler Finsler metrics which are non-Kähler. The examples belong to the unitary invariant complex Randers metrics. Furthermore, a uniformization theorem of the unitary invariant complex Randers metrics with constant holomorphic curvature is proved under the weakly Kähler condition.
Douglas curvature in finsler geometry: A review of the notion and its applications in geometry and physics
本文综述了芬斯勒几何中Douglas曲率的概念、性质、历史发展、特征化、推广及其在几何与物理中的应用，并展望了未来研究方向。
On the Generalized Projective Riemann Curvature in Finsler Geometry
本文研究Finsler几何中广义投影Riemann曲率的性质，提出新的曲率特征化方法，揭示其在投影变换下的不变性及几何意义。
Ricci solitons on Finsler spaces, previously developed by the present authors, are a generalization of Einstein spaces, which can be considered as a solution to the Ricci flow on compact Finsler manifolds. In the present work it is shown that on a Finslerian space, a forward complete shrinking Ricci soliton is compact if and only if it is bounded. Moreover, it is proved that a compact shrinking Finslerian Ricci soliton has finite fundamental group and hence the first de Rham cohomology group vanishes.
We express invariants of Finsler manifolds in a geometrical way by means of using moving planes and their associated Jacobi curves, which are curves in a fixed homogeneous Grassmann manifold. Some applications are given.
A metrical approach to Finsler geometry
本文提出Finsler几何的度量表述，从度量与Finsler联络的相容性公理推导出度量的Hessian性质、非线性联络及Chern或Cartan联络，适用于场论和Finsler引力研究。
In this paper, we explore the similarity between normal homogeneity and $δ$-homogeneity in Finsler geometry. They are both non-negatively curved Finsler spaces. We show that any connected $δ$-homogeneous Finsler space is $G$-$δ$-homo-geneous, for some suitably chosen connected quasi-compact $G$. So $δ$-homogeneous Finsler metrics can be defined by a bi-invariant singular metric on $G$ and submersion, just as normal homogeneous metrics, using a bi-invariant Finsler metric on $G$ instead. More careful analysis shows, in the space of all Finsler metrics on $G/H$, the subset of all $G$-$δ$-homogen…
The $π$-exterior derivative $ød$, which is the Finslerian generalization of the (usual) exterior derivative $d$ of Riemannian geometry, is defined. The notion of a $ød$-closed vector field is introduced and investigated. Various characterizations of $ød$-closed vector fields are established. Some results concerning $ød$-closed vector fields in relation to certain special Finsler spaces are obtained.
A geodesic circle in Finsler geometry is a natural extension of that in a Euclidean space. In this paper, we apply Lie derivatives and the Cartan $Y$-connection to study geodesic circles and (infinitesimal) concircular transformations on a Finsler manifold. We characterize a concircular vector field with some PDEs on the tangent bundle, and then we obtain respective necessary and sufficient conditions for a concircular vector field to be conformal and a conformal vector field to be concircular. We also show conditions for two conformally related Finsler metrics to be concircular, and obtain so…
Horizontal endomorphisms, almost complex structures, vertical, horizontal and complete lifts on prolongation of a Lie algebroid are considered. Then using exact sequences, semisprays are constructed. Moreover, important geometrical objects such as classical distinguished connections, torsions and partial curvatures are studied on prolongation of Lie algebroids. Considering pullback bundle, covariant derivatives are scrutinized based on anchor map. Several Finsler geometry models on Lie algebroid structures, will organized via recent arguments. Finally, it will be overhaulled some special Finsl…
The pullback approach to global Finsler geometry is adopted. Three classes of recurrence in Finsler geometry are introduced and investigated: simple recurrence, Ricci recurrence and concircular recurrence. Each of these classes consists of four types of recurrence. The interrelationships between the different types of recurrence are studied. The generalized concircular recurrence, as a new concept, is singled out.
Studying various functionals and associated gradient ows are known problems in differential geometry. The perpose of this article is to provide a general overview of curvature functionals in Finsler geometry and use their information for introducing different gradient ows on Finsler manifolds.
Here, a non-linear analysis method is applied rather than classical one to study projective changes of Finsler metrics. More intuitively, a projectively invariant pseudo-distance is introduced and characterized with respect to the Ricci tensor and its covariant derivatives.
Let (M.F) be a complete Finsler manifold and P be a minimal and compact submanifold of M. Ric_k(x), x in M is a differential invariant that interpolates between the flag curvature and the Ricci curvature. We prove that if on any geodesic c(t) emanating orthogonally from P we have \int_{0}^{\infty}\mathbf{Ric}_{k}(t)>0, then M is compact.
From radial curvature geometry's standpoint, we prove a sphere theorem of the Grove-Shiohama type for a certain class of compact Finsler manifolds.
Effective field theories with explicit Lorentz violation are intimately linked to Riemann-Finsler geometry. The quadratic single-fermion restriction of the Standard-Model Extension provides a rich source of pseudo-Riemann-Finsler spacetimes and Riemann-Finsler spaces. An example is presented that is constructed from a 1-form coefficient and has Finsler structure complementary to the Randers structure.
The aim of this paper is to develop on the 1-jet space J^1(R,M^3) the Finsler-like geometry (in the sense of distinguished (d-) connection, d-torsions and d-curvatures) of the rheonomic Berwald-Moor metric of order three. Some natural geometrical field theories (gravitational and electromagnetic) produced by the preceding rheonomic Berwald-Moor metric of order three are also exposed.
The global study of Riemannian-Finsler geometry
本文比较研究Riemannian与Finsler几何，探讨切点和共轭区域的差异，分析凸函数在Riemannian流形上的拓扑与微分结构及Finsler流形中的类比结果。
In this paper, we identify the Bott connection on the natural foliation of the projective sphere bundle of a Finsler manifold to the Chern connection of this manifold. As a consequence, the symmetrization of the Bott connection turns out to be the Cartan connection of the Finsler manifold. Following Liu-Zhang \cite{LiuZ}, the Cartan connection can also be obtained through an adiabatic limit process. Furthermore, a Chern-Simons type form is defined and its conformal properties are discussed.
Finsler Geometry, Spacetime & Gravity -- From Metrizability of Berwald Spaces to Exact Vacuum Solutions in Finsler Gravity
本文研究了Finsler几何中伯瓦尔德空间的可度量性问题，并探讨了Finsler引力中的精确真空解和引力波观测特征。
The paper develops the Finsler-like geometry on the 1-jet space for the jet conformal Minkowski (JCM) metric, which naturally extends the Minkowski metric in the Chernov-Pavlov framework. To this aim there are determined the nonlinear connection, distinguished (d-) Cartan linear connection, d-torsions and d-curvatures. The field geometrical gravitational and electromagnetic d-models based on the JCM metric are discussed.
We show how to compute tensor derivatives and curvature tensors using affine connections. This allows for all computations to be obtained without using coordinate systems, in a way that parallels the computations appearing in classical Riemannian Geometry. In particular, we obtain Bianchi identities for the curvature tensor of any anisotropic connection, we compare the curvature tensors of any two anisotropic connections, and we find a family of anisotropic connections which are well suited to study the geometry of Finsler metrics.
Bipartite Riemann-Finsler geometries with complementary Finsler structures are constructed. Calculable examples are presented based on a bilinear-form coefficient for explicit Lorentz violation.
The aim of this article is to establish a Toponogov type triangle comparison theorem for Finsler manifolds, in the manner of radial curvature geometry. We consider the situation that the radial flag curvature is bounded below by the radial curvature function of a non-compact surface of revolution, the edge opposite to the base point is contained in a Berwald-like region, and that the Finsler metric is convex enough in the radial directions in that region.
The physics of classical particles in a Lorentz-breaking spacetime has numerous features resembling the properties of Finsler geometry. In particular, the Lagrange function plays a role similar to that of a Finsler structure function. A summary is presented of recent results, including new calculable Finsler structures based on Lagrange functions appearing in the Lorentz-violation framework known as the Standard-Model Extension.
Positive solutions of a critical equation in sub-Finsler geometry
本文针对 Finsler 与次 Riemann 几何交汇处的非线性算子，给出临界 Yamabe 型方程的双参数显式正解族。
Here, by extending the definition of circle to Finsler geometry, we show that, every circle-preserving local diffeomorphism is conformal. This result implies that in Finsler geometry, the definition of concircular change of metrics, a priori, does not require the conformal assumption.
The natural reductivity in Finsler geometry in terms of geodesic graphs
本文提出自然还原Finsler流形的测地线图新定义，构造纯Finsler自然还原度量例子，并描述非自然还原度量的测地线图及单形式影响。
In this paper we prove several multiplicity results of $t$-periodic light rays in conformally stationary spacetimes using the Fermat metric and the extensions of the classical theorems of Gromoll-Meyer and Bangert-Hingston to Finsler manifolds. Moreover, we exhibit some stationary spacetimes with a finite number of $t$-periodic light rays and compute a lower bound for the period of the light rays when the flag curvature of the Fermat metric is $η$-pinched.
Using a navigation process with the datum $(F,V)$, in which $F$ is a Finsler metric and the smooth tangent vector field $V$ satisfies $F(-V(x))>1$ everywhere, a Lorentz Finsler metric $\tilde{F}$ can be induced. Isoparametric functions and isoparametric hypersurfaces with or without involving a smooth measure can be defined for $\tilde{F}$. When the vector field $V$ in the navigation datum is homothetic, we prove the local correspondences between isoparametric functions and isoparametric hypersurfaces before and after this navigation process. Using these correspondences, we provide some exampl…
Harmonic vector fields and the Hodge Laplacian operator on Finsler geometry
本文在 Finsler 流形上定义了水平微分、散度与 $p$-调和形式，证明了 Hodge 型定理，提出了调和向量场的新定义，并导出了基于调和 Ricci 标量的 Bochner-Yano 型分类定理。
We determine a 2-codimensional CR-structure on the slit tangent bundle $T_0M$ of a Finsler manifold $(M, F)$ by imposing a condition regarding the almost complex structure $Ψ$ associated to $F$ when restricted to the structural distribution of a framed $f$-structure. This condition is satisfied when $(M, F)$ is of scalar flag curvature (particularly flat) and in the Riemannian case $(M, g)$ this last condition means that $g$ is of constant curvature. This CR-structure is finally generalized by using one positive number but under more difficult conditions.
In Finsler geometry, we use calculus to study the geometry of regular inner metric spaces. In this note I will briefly discuss various curvatures and their geometric meanings from the metric geometry point of view, without going into the forest of tensors.
The book "Handbook of Finsler geometry" has been included with a CD containing an elegant Maple package, FINSLER, for calculations in Finsler geometry. Using this package, an example concerning a Finsler generalization of Einstein's vacuum field equations was treated. In this example, the calculation of the components of the hv-curvature of Cartan connection leads to wrong expressions. On the other hand, the FINSLER package works only in dimension four. We introduce a new Finsler package in which we fix the two problems and solve them. Moreover, we extend this package to compute not only the g…
We investigate existence and uniqueness of p-means and the median of a probability measure on a Finsler manifold, in relation with the convexity of the support of the measure. We prove that the p-mean is the limit point of a continuous time gradient flow. Under some additional condition which is always satisfied for larger than or equal to 2, a discretization of this path converges to the p-mean. This provides an algorithm for determining those Finsler center points.
In 1929, Paul Funk and Ludwig Berwald gave a characterization of Hilbert geometries from the Finslerian viewpoint. They showed that a smooth Finsler metric in a convex bounded domain of $\mathbb{R}^n$ is the Hilbert geometry in that domain if and only if it is complete, if its geodesics are straight lines and if its flag curvature is equal to -1. The goal of this chapter is to explain these notions in details, to illustrate the relation between Hilbert geometry, Finsler geometry and the calculus of variations, and to prove the Funk-Berwald characterization Theorem.
Stable recovery of a simple irreversible Finsler geometry from travel time data
针对不可逆芬斯勒几何，引入适应不可逆度量空间的Gromov-Hausdorff距离，证明在简单性条件下走时数据可唯一且Lipschitz稳定地恢复几何结构。
In this paper, we introduce the weighted projective Ricci curvature as an extension of projective Ricci curvature introduced by Z. Shen. We characterize the class of Randers metrics of weighted projective Ricci flat curvature. We find the necessary and sufficient condition under which a Kropina metric has weighted projective Ricci flat curvature. Finally, we show that every projectively flat metric with isotropic weighted projective Ricci and isotropic S-curvature is a Kropina metric or Randers metric.
In the present paper, we introduce and investigate the notion of a semi concurrent vector field on a Finsler manifold. We show that some special Finsler manifolds admitting such vector fields turn out to be Riemannian. We prove that Tachibana's characterization of Finsler manifolds admitting a concurrent vector field leads to Riemannain metrics. We give an answer to the question raised in \cite{DWF}: "Is any n-dimensional Finsler manifold $(M,F)$, admitting a non-constant smooth function $f$ on $M$ such that $\frac{\partial f}{\partial x^i}\frac{\partial g^{ij}}{\partial y^k}=0$, a Riemannian …
Here, a Finsler manifold (M, F) is considered with corresponding curvature tensor, regarded as 2-forms on the bundle of non-zero tangent vectors. Certain subspaces of the tangent spaces of M determined by the curvature are introduced and called k-nullity foliations of the curvature operator. It is shown that if the dimension of foliation is constant then the distribution is involutive and each maximal integral manifold is totally geodesic. Characterization of the k-nullity foliation is given, as well as some results concerning constancy of the flag curvature, and completeness of their integral…
On the foundations and applications of Lorentz-Finsler Geometry
本文探讨了洛伦兹-芬斯勒几何在相对论物理及其他领域的应用，介绍了其基础概念、关键应用及未来发展方向，包括几何结构、与黎曼几何的联系以及在经典力学中的波传播应用。
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。