Stefan Klempnauer
We use a theorem of P. Berger and D. Turaev to construct an example of a Finsler geodesic flow on the 2-torus with a transverse section, such that its Poincaré return map has positive metric entropy. The Finsler metric generating the flow can be chosen to be arbitrarily $C^\infty$-close to a flat metric.
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Uniqueness of shortest closed geodesics for generic Finsler metrics
在闭流形上的Finsler（或黎曼）度量的每一个共形类中，存在Finsler度量的一个剩余子集，使得关于剩余Finsler度量，在自由回路的任何非平凡同伦类中，恰好存在一个最短的测地线环路.
In this paper, we study the existence and multiplicity problems for orthogonal Finsler geodesic chords in a manifold with boundary which is homeomorphic to a N-dimensional disk. Under a suitable assumption, which is weaker than convexity, we prove that, if the Finsler metric is reversible, then there are at least N orthogonal Finsler geodesic chords that are geometrically distinct. If the reversibility assumption does not hold, then there are at least two orthogonal Finsler geodesic chords with different values of the energy functional.
If all prime closed geodesics on $(S^n,F)$ with an irreversible Finsler metric $F$ are irrationally elliptic, there exist either exactly $2\left[\frac{n+1}{2}\right]$ or infinitely many distinct closed geodesics. As an application, we show the existence of three distinct closed geodesics on bumpy Finsler $(S^3,F)$ if any prime closed geodesic has non-zero Morse index.
In this paper, we prove that for every Finsler $n$-dimensional sphere $(S^n,F), n\ge 3$ with reversibility $λ$ and flag curvature $K$ satisfying $\left(\fracλ{1+λ}\right)^2<K\le 1$, there exist at least three distinct closed geodesics and at least two of them are elliptic if the number of prime closed geodesics is finite. When $n\ge 6$, these three distinct closed geodesics are non-hyperbolic.
It's well known that the n-sphere $S^n$ is the universal double covering of the $n$-dimensional real projective space $\mathbb{R}P^n$ and then any Finsler metric on $\mathbb{R}P^n$ induces a Finsler metric of $S^n$. In this paper, we prove that for every Finsler $(S^n, F)$ for $n\geq3$ whose metric is induced by irreversible Finsler $(\mathbb{R}P^n,F)$ with reversibility $λ$ and flag curvature $K$ satisfying $(\fracλ{λ+1})^2<K\leq 1$, there exist at least $n-1$ prime closed geodesics on $(S^n, F)$. Furthermore, if there exist finitely many distinct closed geodesics on $(S^n, F)$, then there ex…
We show that a small perturbation of the boundary distance function of a simple Finsler metric on the $n$-disc is also the boundary distance function of some Finsler metric. (Simple metric form an open class containing all flat metrics.) The lens map is map that sends the exit vector to the entry vector as a geodesic crosses the disc. We show that a small perturbation of a lens map of a simple Finsler metric is in its turn the lens map of some Finsler metric. We use this result to construct a smooth perturbation of the metric on the standard 4-dimensional sphere to produce positive metric entr…
We prove some rigidity results on geodesic orbit Finsler spaces with non-positive curvature. In particular, we show that a geodesic Finsler space with strictly negative flag curvature must be a non-compact Riemannian symmetric space of rank one.
Multiple closed geodesics on Finsler $3$-dimensional sphere
本文在所有素闭测地线Morse指数非零的条件下，证明了Anosov关于bumpy Finsler $S^3$至少存在四条素闭测地线的猜想。
In this paper, we establish first the resonance identity for non-contractible homologically visible prime closed geodesics on Finsler $n$-dimensional real projective space $(\mathbb{R}P^n,F)$ when there exist only finitely many distinct non-contractible closed geodesics on $(\mathbb{R}P^n,F)$, where the integer $n\geq2$. Then as an application of this resonance identity, we prove the existence of at least two distinct non-contractible closed geodesics on $\mathbb{R}P^{n}$ with a bumpy and irreversible Finsler metric. Together with two previous results on bumpy and reversible Finsler metrics in…
We study singularities of geodesics flows in two-dimensional generalized Finsler spaces (pseudo-Finsler spaces). Geodesics are defined as extremals of a certain auxiliary functional whose non-isotropic extremals coincide with extremals of the action functional. This allows to consider isotropic lines as (unparametrized) geodesics.
Let $M$ be a compact simply connected manifold satisfying $H^*(M;\mathbf{Q})\cong T_{d,n+1}(x)$ for integers $d\ge 2$ and $n\ge 1$. If all prime closed geodesics on $(M,F)$ with an irreversible bumpy Finsler metric $F$ are elliptic, either there exist exactly $\frac{dn(n+1)}{2}$ (when $d\ge 2$ is even) or $(d+1)$ (when $d\ge 3$ is odd) distinct closed geodesics, or there exist infinitely many distinct closed geodesics.
Homogeneous geodesics in homogeneous Finsler spaces
本文研究了齐性Finsler空间中的齐性测地线。我们首先给出一个简单的标准，特点测地线向量。证明了李群上相对于双不变Finsler度量的测地线是单参数子群的陪集。建立了紧半单李群上无穷多个齐次测地线的存在性。我们引入了自然约化齐性Finsler空间的概念。作为特例，我们研究齐性Randers空间中的齐性测地线。最后，我们研究了齐次测地线的曲率性质。特别地，我们证明了S-曲率沿齐次测地线为零。
In this paper, we study geodesics and geodesic vectors for homogeneous exponential Finsler space and homogeneous infinite series Finsler space. Further, we find necessary and sufficient condition for a non-zero vector in these homogeneous spaces to be a geodesic vector.
Homogeneous geodesics of left invariant Finsler metrics
本文研究了李群上左不变Finsler度量的齐次测地线集。我们首先给出一个简单的标准，特点测地线向量。作为应用，我们研究了李群上双不变Finsler度量的一些几何性质.特别地，给出了左不变Randers度量是Berwald型的一个充要条件.最后给出了齐次测地线与限制Finsler度量的临界点的对应关系。从而得到了关于齐次测地线存在性的结果。
The second closed geodesic on Finsler spheres of dimension n>2
证明了n维球面上存在至少两条几何上不同的闭测地线，其中n>2，Finsler度量是不可逆的。
Remarks on magnetic flows and magnetic billiards, Finsler metrics and a magnetic analog of Hilbert's fourth problem
我们解释磁性台球芬斯勒的，并描述了一个类似的磁性台球的字符串建设。描述了“入射角等于反射角”定律的芬斯勒台球。我们刻画了平面上的Finsler度量，其测地线是一个固定半径的圆。这是一个磁性模拟希尔伯特的第四个问题，要求描述芬斯勒度量的测地线是直线。
In this paper, we prove that for every irreversible Finsler $n$-dimensional real projective space $(\mathbb{R}P^n,F)$ with reversibility $λ$ and flag curvature $K$ satisfying $\frac{16}{9}\left(\fracλ{1+λ}\right)^2<K\le 1$ with $λ<3$, there exist at least $n-1$ non-contractible closed geodesics. In addition, if the metric $F$ is bumpy with $\frac{64}{25}\left(\fracλ{1+λ}\right)^2<K\le 1$ and $λ<\frac{5}{3}$, then there exist at least $2[\frac{n+1}{2}]$ non-contractible closed geodesics, which is the optimal lower bound due to Katok's example. The main ingredients of the proofs are the Fadell-R…
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 $Sp(n)/Sp(n-1)$.
Multiplicity and stability of closed geodesics on bumpy Finsler 3-spheres
本文证明了对于每一个具有凹凸不可逆度量$F$的$\Q$ -同调Finsler 3-球面$(M,F)$，要么存在两条非双曲素闭测地线，要么至少存在三条素闭测地线.
Existence of closed geodesics on positively curved Finsler manifolds
对于球面和射影空间上具有正旗曲率的不可逆Finsler度量，我们给出了关于闭测地线的个数和长度以及它们的稳定性的结果.
We survey some results on the existence (and non-existence) of periodic Reeb orbits on contact manifolds, both in the open and closed case. We place these statements in the context of Finsler geometry by including a proof of the folklore theorem that the Finsler geodesic flow can be interpreted as a Reeb flow. As a mild extension of previous results we present existence statements on periodic Reeb orbits on contact manifolds with suitable supporting open books.
Multiple closed geodesics on bumpy Finsler $n$-spheres
本文证明了：对于任意有理同调$n$维球面$S^n$上的任意凹凸Finsler度量$F$，总存在至少两条不同的素闭测地线.
In this paper, we consider a Finsler space with a Randers change of Quartic metric F = $\sqrt[4]{α^4 + β^4} + β$. The conditions for this space to be with reversible geodesics are obtained. Further, we study some geometrical properties of F with reversible geodesics and prove that the Finsler metric F induces a generalized weighted quasi-distance $d_F$ on M.
Finsler surfaces with prescribed geodesics
我们构造所有的Finsler度量的两个球的测地线是圆，并表明任何（可逆）的路径几何的二维流形上的局部系统的测地线的Finsler度量。
Geodesically reversible Finsler 2-spheres of constant curvature
一个Finsler空间被称为测地线可逆的，如果每个定向测地线都可以被重新参数化为一个具有相反定向的测地线。一个可逆的芬斯勒空间是测地线可逆的，但反过来不一定成立。本文基于LeBrun和Mason的最新工作，证明了2-球面上具有常旗曲率的测地线可逆Finsler度量必然是射影平坦的.作为推论，利用作者以前的一个结果，证明了2-球面上一个可逆的常旗曲率Finsler度量必然是一个常Gauss曲率的Riemann度量，从而解决了Finsler几何中一个长期存在的问题.
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.
Geodesic graphs for geodesic orbit Finsler $(α,β)$ metrics on spheres
研究球面上由 Riemannian 测地轨道度量生成的不变 Finsler $(α,β)$ 度量，通过显式测地图构造刻画其轨道条件，并证明非实射影空间不存在纯 Finsler 情形。
In a recent paper, it was claimed that any homogeneous Finsler space of odd dimension admits a homogeneous geodesic through any point. For the proof, the algebraic method dealing with the reductive decomposition of the Lie algebra of the isometry group was used. However, the proof contains a serious gap. In the present paper, homogeneous geodesics in Finsler homogeneous spaces are studied using the affine method, which was developed in earlier papers by the author. The mentioned statement is proved correctly and it is further proved that any homogeneous Berwald space or homogeneous reversible …
In this paper, we prove that for every Finsler $4$-dimensional sphere $(S^4,F)$ with reversibility $λ$ and flag curvature $K$ satisfying $\frac{25}{9}\left(\fracλ{1+λ}\right)^2<K\le 1$ with $λ<\frac{3}{2}$, either there exist at least four prime closed geodesics, or there exist exactly three prime non-hyperbolic closed geodesics and at least two of them are irrationally elliptic.
On the average indices of closed geodesics on positively curved Finsler spheres
本文证明了：在任意Finsler $n$ -球面$(S^n, F)$上，对于$n\ge 6$，如果可逆性$λ$和旗曲率$K$满足$(\fracλ{λ+1})^2<K\le 1$，则存在无穷多条素闭测地线，或者存在具有无理平均指数的$[\frac{n}{2}]-2$闭测地线.如果度量是不平坦的，则当闭测地线的个数有限时，存在具有无理平均指数的$n-3$闭测地线.
The index growth and multiplicity of closed geodesics
在最近的论文\cite{LoD1}中，我们将Finsler流形上的闭测地线分为有理和无理两类，并对有理闭测地线迭代的指数增长性质给出了完整的理解.本文证明了有理闭测地线不可能是每个不可逆或可逆（包括黎曼）Finsler球面上唯一的闭测地线，并且证明了每个紧致单连通不可逆或可逆（包括黎曼）Finsler三维流形上至少存在两条不同的闭测地线.本文研究了Finsler流形上无理闭测地线的指数增长性质。本文的研究使我们能够将\cite{LoD1}关于有理的结果，\cite{DuL1}，\cite{Rad4}和\cite{Rad5}关于球面上完全非退化闭测地线的结果以及$\CP^2$的结果推广到每一个紧致单连通Finsler流形上.然后证明了在每个紧致单连通的不可逆或可逆（包括黎曼）Finsler四维流形上至少存在两条不同的闭测地线。
We extend two celebrated theorems on closed geodesics of Riemannian 2-spheres to the larger class of reversible Finsler 2-spheres: Lusternik-Schnirelmann's theorem asserting the existence of three simple closed geodesics, and Bangert-Franks-Hingston's theorem asserting the existence of infinitely many closed geodesics. In order to prove the first theorem, we employ the generalization of Grayson's curve shortening flow developed by Angenent-Oaks.
Let $M=S^n/ Γ$ and $h$ be a nontrivial element of finite order $p$ in $π_1(M)$, where the integer $n, p\geq2$, $Γ$ is a finite abelian group which acts freely and isometrically on the $n$-sphere and therefore $M$ is diffeomorphic to a compact space form. In this paper, we prove that for every irreversible Finsler compact space form $(M,F)$ with reversibility $λ$ and flag curvature $K$ satisfying \[ \frac{4p^2}{(p+1)^2} \big(\fracλ{λ+1} \big)^2 < K \leq 1,\;\;λ< \frac{p+1}{p-1}, \] there exist at least $n-1$ non-contractible closed geodesics of class $[h]$. In addition, if the metric $F$ is bum…
We prove that a reversible Berwaldian Finsler structure $\left(M,F\right)$ with base-independent non-negative radial flag curvature and large volume growth does not have any closed geodesics.
We study non-reversible Finsler metrics with constant flag curvature 1 on S^2 and show that the geodesic flow of every such metric is conjugate to that of one of Katok's examples, which form a 1-parameter family. In particular, the length of the shortest closed geodesic is a complete invariant of the geodesic flow. We also show, in any dimension, that the geodesic flow of a Finsler metrics with constant positive flag curvature is completely integrable. Finally, we give an example of a Finsler metric on~$S^2$ with positive flag curvature such that no two closed geodesics intersect and show that…
Equidistributed Closed Geodesics on Closed Finsler and Riemannian Surfaces
本文利用嵌入切触同调的体积性质、局部变分构造与横截性论证，证明了闭曲面上一般Finsler、对称Finsler及Riemann度量存在非退化闭测地线的等分布序列。
A Finsler space $(M,F)$ is called a geodesic orbit space if any geodesic of constant speed is the orbit of a one-parameter subgroup of isometries of $(M, F)$. In this paper, we study Finsler metrics on Euclidean spaces which are geodesic orbit metrics. We will show that, in this case $(M, F)$ is a fiber bundle over a symmetric Finsler space $M_1$ of non-compact type such that each fiber $M_2$ is a totally geodesic nilmanifold with a step-size at most 2, and the projection $π:M\rightarrow M_1$ is a Finslerian submersion. Furthermore, when $M_1$ has no Hermitian symmetric factors, the fiber bund…
For compact manifolds with infinite fundamental group we present sufficient topological or metric conditions ensuring the existence of two geometrically distinct closed geodesics. We also show how results about generic Riemannian metrics can be carried over to Finsler metrics.
Two-step homogeneous geodesics in some homogeneous Finsler manifolds
本文探讨了同质Finsler空间中两步Finsler测地轨道空间的条件及示例，扩展了同质Riemannian空间中同质测地线的概念。
Existence of closed geodesics on Finsler $n$-spheres
本文证明了：在任何可逆性$λ$满足$F^2<(\frac{λ+1}λ)^2g_0$和$l(S^n, F)\ge π(1+\frac{1}λ)$的Finsler $n$ -球面$(S^n, F)$上，总存在至少一条不自交的$n$素闭测地线，其中$g_0$是$S^n$上常曲率为1的标准黎曼度量，$l(S^n, F)$是$(S^n, F)$上最短测地线环的长度.我们还研究了这些闭测地线的稳定性。
Morse theory of causal geodesics in a stationary spacetime via Morse theory of geodesics of a Finsler metric
我们证明了共形标准定态时空中的类光测地线的指数等于其作为与该时空相关联的芬斯勒度量的测地线的空间投影的指数。利用相应的Finsler流形上的莫尔斯理论，得到了标准类时Killing向量场的积分线与点之间的类光测地线的莫尔斯关系.为此，我们证明了Finsler度量的能量泛函的分裂引理。最后，我们表明，减少到莫尔斯理论的芬斯勒流形也可以做类时测地线。
Closed geodesics on positively curved Finsler spheres
本文证明了：对于任意Finsler $n$ -球面$(S^n, F)$，对于$n\ge 3$，如果可逆性$λ$和旗曲率$K$满足$(\fracλ{λ+1})^2<K\le 1$，则存在无穷多条素闭测地线，或者存在一条椭圆闭测地线，其线性化Poincaré映射至少有一个特征值为带无理$μ$的$\exp(πi μ)$形式.而且，在满足上述拼挤条件的任何$(S^3, F)$上，总存在三条素闭测地线.
Structure of geodesics for Finsler metrics arising from Riemannian g.o. metrics
本文研究由多个黎曼测地轨道度量导出的齐性Finsler度量的测地线，证明正相关黎曼测地轨道度量导出的Finsler度量具测地轨道性质，还分析了S⁷上新型(α₁,α₂,α₃)Finsler度量的测地图，补全了此前球面测地轨道度量分类的缺失情形。
When a closed Finsler manifold admits continuous isometric actions, estimating the number of orbits of prime closed geodesics seems a more reasonable substitution for estimating the number of prime closed geodesics. To generalize the works of H. Duan, Y. Long, H.B. Rademacher, W. Wang and others on the existence of two prime closed geodesics to the equivariant situation, we purpose the question if a closed Finsler manifold has only one orbit of prime closed geodesic if and only if it is a compact rank-one Riemannian symmetric space. In this paper, we study this problem in homogeneous Finsler g…
Ergodic components and topological entropy in geodesic flows of surfaces
考虑2-球面和2-环面上可逆Finsler度量的测地流，其测地流的拓扑熵为零.在A. Katok，我们讨论了芬斯勒度量的例子，这两个表面上，有大遍历组件的测地流的单位切丛。另一方面，利用J. Franks和M. Handel的方法，我们证明了当Finsler度量具有正的旗曲率和至少两条闭测地线时，2-球面的满单位切丛中不存在遍历性和稠密轨道.在2-环面的情况下，我们证明了遍历性被限制到严格的子集之间的流动不变环面的2-环面的单位切丛的管。
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.
In this note, we prove that given a submanifold $P$ in a Finsler manifold $(M,F)$, (i) the orthogonal geodesics to $P$ minimize the distance from $P$ at least in some interval, (ii) there exist tubular neighbourhoods around each point of $P$, (iii) the distance from $P$ is smooth in some open neighbourhood of $P$ (but not necessarily in $P$).
We show that the geodesic ray transform is injective on scalar functions on spherically symmetric reversible Finsler manifolds where the Finsler norm satisfies a Herglotz condition. We use angular Fourier series to reduce the injectivity problem to the invertibility of generalized Abel transforms and by Taylor expansions of geodesics we show that these Abel transforms are injective. Our result has applications in linearized boundary rigidity problem on Finsler manifolds and especially in linearized elastic travel time tomography.
In this paper, we prove that for every Finsler $n$-dimensional sphere $(S^{n},F)$ with reversibility $\lm$ and flag curvature $K$ satisfying $\left(\frac{\lm}{1+\lm}\right)^2<K\le 1$, either there exist infinitely many closed geodesics, or there exist at least two elliptic closed geodesics and each linearized Poincaré map has at least one eigenvalue of the form $e^{\sqrt{-1}þ}$ with $þ$ being an irrational multiple of $π$.
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 that on every Finsler manifold $(M,\,F)$ with reversibility $λ$ and flag curvature $K$ satisfying $\left(\fracλ{λ+1}\right)^2<K\le 1$, there exist $[\frac{\dim M+1}{2}]$ closed geodesics. If the number of closed geodesics is finite, then there exist $[\frac{\dim M}{2}]$ non-hyperbolic closed geodesics. Moreover, there are 3 closed geodesics on $(M,\,F)$ satisfying the above pinching condition when $\dim M=3$.
In this paper, we discuss how to travel along horizontal broken geodesics of a homogenous Finsler submersion, i.e., we study, what in Riemannian geometry was called by Wilking, the dual leaves. More precisely, we investigate the attainable sets $\mathcal{A}_{q}(\mathcal{C})$ of the set of analytic vector fields $\mathcal{C}$ determined by the family of horizontal unit geodesic vector fields $\mathcal{C}$ to the fibers $\mathcal{F}=\{ρ^{-1}(c)\}$ of a homogenous analytic Finsler submersion $ρ: M\to B$. Since reverse of geodesics don't need to be geodesics in Finsler geometry, one can have examp…
Biharmonic curves in Finsler spaces
双调和曲线是测地线的推广，在弹性理论和计算机科学的各个分支中都有应用。本文首次研究了Finsler几何空间中的双调和曲线，主要内容包括：双调和曲线方程的推导，几类Finsler空间中非测地双调和曲线的存在性及其性质。我们证明，例如，测地曲率是常数沿芬斯勒双调和曲线。对两个具体的Finsler度量给出了双调和方程的积分.
A class of Finsler surfaces whose geodesics are circles
我们确定所有Finsler度量的Randers类型的黎曼部分是一个标量倍的欧几里德度量，在一个开子集的欧几里德平面，其测地线是圆。我们证明了黎曼部分必须是恒定的高斯曲率，并且对于每个这样的黎曼度量有一类Randers度量满足条件，确定到增加一个总导数，取决于一个单一的参数。作为几个应用之一，我们展示了芬斯勒度量，其测地线是庞加莱盘中两个可能方向中的定向等环。
In this paper, we prove that lightlike geodesics of a pseudo-Finsler manifold and its focal points are preserved up to reparametrization by anisotropic conformal changes, using the Chern connection and the anisotropic calculus and the fact that geodesics are critical points of the energy functional and Jacobi fields, the kernel of its index form. This result has applications to the study of Finsler spacetimes.
We extend the notion of the geometric entropy of foliation to foliated manifolds equipped with leafwise Finsler structure. We study the relation between the geometric entropy and the topological entropy of the holonomy pseudogroup. The case of foliated manifold with leafwise Randers structure. In this case the estimates for one dimensional foliation defined by a vector field in term of topological entropy of a flow are presented.
Finsler and sub-Finsler geodesics with chattering
本文构造了 Finsler 和 sub-Finsler 流形及 Carnot 群上的左不变结构，证明其测地线存在抖振现象，从而否定回答了 Le Donne 问题，并给出了极值发生抖振的充分条件。
A Finsler metric is geodesically reversible if geodesics remain geodesics after a change of orientation. Asymmetric norms on vector spaces and Funk metrics in the interior of convex bodies are examples of geodesically reversible metrics that are not necessarily sums of reversible metrics and closed 1-forms. However, there seem to be few such examples in closed manifolds. In this paper the theory of volumes and areas on Finsler spaces is applied to establish a number of rigidity theorems which partially explain this paucity of examples. These rigidity results settle some hitherto unsolved cases…
Let $M=S^{2n+1}/ Γ$, $Γ$ is a finite group which acts freely and isometrically on the $(2n+1)$-sphere and therefore $M$ is diffeomorphic to a compact space form. In this paper, we first investigate Katok's famous example about irreversible Finsler metrics on the spheres to study the topological structure of the contractible component of the free loop space on the compact space form $M$, then we apply the result to establish the resonance identity for homologically visible contractible minimal closed geodesics on every Finsler compact space form $(M, F)$ when there exist only finitely many dist…
Two closed geodesics on compact bumpy Finsler manifolds
本文证明了具有有限基本群的紧致颠簸Finsler流形上至少存在两条闭测地线，并指出该下界可由Katok 2-球面和2-实射影空间达到。
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