David Csaba Kertesz
A Finsler function $F$ is affinely rigid if its canonical spray is uniquely metrizable, in the sense that if $\bar F$ is another Finsler function whose canonical spray is $S$, then $d(F/\bar F)=0$. In this short note we explore some sufficient conditions for a Finsler function to be affinely rigid, and discuss open problems.
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Strong rigidity of constant curvature Finsler manifolds
本文将Obata-Tanno定理推广到Finsler几何中，得到了如下刚性结果：每一个具有正常旗曲率的完备连通Finsler流形等距同胚于一个具有一定Finsler度量的$n$ -球面，反之亦然.
We provide an extension of Obata's theorem to Finsler geometry and establish some rigidity results based on a second order differential equation. Mainly, we prove that every complete connected Finsler manifold of positive constant flag curvature is isometrically homeomorphic to an Euclidean sphere endowed with a certain Finsler metric and vice versa. Based on these results, we present a classification of Finsler manifolds which admit a transnormal function. Specifically, we show that if a complete Finsler manifold admits a transnormal function with exactly two critical points, then it is homeo…
We investigate the rigidity problem for the sharp spectral gap on Finsler manifolds of weighted Ricci curvature bound $\text{Ric}_{\infty} \geq K > 0$. Our main results show that if the equality holds, the manifold necessarily admits a diffeomorphic splitting (or isometric splitting in the particular class of Berwald spaces). This splitting phenomenon is comparable to the Cheeger-Gromoll type splitting theorem by Ohta. We also obtain the rigidity results of logarithmic Sobolev and Bakry-Ledoux isoperimetric inequalities via needle decomposition as corollaries.
On rigidity of Finsler manifolds without conjugate points and with constant $S$-curvature
本文研究无共轭点且具常S-曲率的闭芬斯勒流形的刚性，证明满足特定条件的闭C^ω或C^∞芬斯勒流形必为黎曼流形，进而得到相关刚性结果，方法基于Cartan向量场与Green丛几何的分析。
Projective and Finsler metrizability: parameterization-rigidity of the geodesics
在这项工作中，我们表明，对于Finsler函数$F$的测地线喷雾$S$的最自然的投影变形$\widetilde{S}=S -2 λF\mathbb C$导致非Finsler度量化喷雾，几乎每一个值的$λ\in \mathbb R$。这一结果表明，如何刚性的度量化属性相对于某些重新参数化的测地线。作为结果，我们得到的投影类的任意喷雾包含无限多的喷雾，芬斯勒度量化。
Scattering rigidity for Hamiltonian systems with an application to Finsler geometry
本文研究了Hamiltonian系统中的散射刚性，通过散射关系反演Hamiltonian函数，并应用于非陷阱Finsler流形的半全局透镜刚性证明。
Suppose that $M$ is a Kähler manifold with a pole such that its holomorphic sectional curvature is bounded from below by a constant and its radial sectional curvature is also bounded from below. Suppose that $N$ is a strongly pseudoconvex complex Finsler manifold such that its holomorphic sectional curvature is bounded from above by a negative constant. In this paper, we establish a Schwarz lemma for holomorphic mappings $f$ form $M$ into $N$. As applications, we obtain a Liouville type rigidity result for holomorphic mappings $f$ from $M$ into $N$, as well as a rigidity theorem for bimeromorp…
Entropy collapse versus entropy rigidity for Reeb and Finsler flows
该研究对比切触流形上Reeb流的拓扑熵可任意小的熵坍缩现象，与许多闭流形上归一化Finsler测地流拓扑熵有一致正下界的熵刚性性质，揭示两类流的熵行为差异。
Conformal boundary rigidity for simple Finsler metrics
本文证明了简单芬斯勒流形在共形因子下具有稳定性，通过刻画边界距离函数的奇异性定义$H^2$范数，并利用改编的Mukhometov分部积分技术获得Sobolev范数与$L^2$范数之间的稳定性估计。
Consider a smooth manifold $M$ equipped with a bracket generating distribution $D$. Two sub-Riemannian metrics on $(M,D)$ are said to be projectively (resp. affinely) equivalent if they have the same geodesics up to reparameterization (resp. up to affine reparameterization). A sub-Riemannian metric $g$ is called rigid (resp. conformally rigid) with respect to projective/affine equivalence, if any sub-Riemannian metric which is projectively/affinely equivalent to $g$ is constantly proportional to $g$ (resp. conformal to $g$). In the Riemannian case the local classification of projectively and a…
On Negatively Curved Finsler Manifolds of Scalar Curvature
本文证明了维数n>2的紧致流形上负曲Finsler度量的一个整体刚性定理。证明了对于这样的Finsler流形，如果旗曲率是切丛上的标量函数，则Finsler度量是Randers型的.我们还研究了Finsler度量是局部射影平坦的情况。
Isoperimetric inequality for Finsler manifolds with non-negative Ricci curvature
本文证明了具有非负 Ricci 曲率和 Euclidean 体积增长的测度 Finsler 流形的尖锐等周不等式，并在附加假设下证明了其刚性结果，进而推导出 Euclidean 空间中锥的加权各向异性等周不等式在不可逆情形下的刚性。
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.
Finsler Manifolds with Nonpositive Flag Curvature and Constant S-curvature
旗曲率是黎曼几何中截面曲率的自然扩张，S-曲率是一个非黎曼量，对于黎曼度量，S-曲率为零. R^n中具有负旗曲率和常S曲率的开子集上存在（不完全）非黎曼Finsler度量。本文证明了一个整体刚性定理，即如果流形是紧的，则每个具有负旗曲率和常S曲率的Finsler度量必是黎曼度量。我们还研究了非正旗曲率的情况。
Normal families of holomorphic mappings between complex Finsler manifolds
本文证明复 Finsler 流形间全纯映射的 Montel 型正规族定理，并据此推广强负曲率族结果，得到 Cartan-Carathéodory-Kaup-Wu 定理等 Finsler 版本。
Unexpected Analytic Phenomena on Finsler Manifolds
本文研究平坦Finsler Cartan-Hadamard流形上的分析性质，发现Nash嵌入定理失效、Sobolev空间非线性、Hardy和不确定性不等式完全失效，而CKN不等式存在参数依赖的尖锐阈值，并归因于S-曲率下界。
Geodesics, which play an important role in spray-Finsler geometry, are integral curves of a spray vector field on a manifold. Some comparison theorems and rigidity issues are established on the completeness of geodesics of a spray or a Finsler metric. In this paper, projectively flat sprays with weak Ricci constant (eps. constant curvature) are classified at the level of geodesics. Further, a geodesic method is introduced to determine an $n$-dimensional spray based on a family of curves with $2(n-1)$ free constant parameters as geodesics. Finally, it shows that a spray is projectively complete…
In this paper we study the invariant metrizability and projective metrizability problems for the special case of the geodesic spray associated to the canonical connection of a Lie group. We prove that such canonical spray is projectively Finsler metrizable if and only if it is Riemann metrizable. This result means that this structure is rigid in the sense that considering left-invariant metrics, the potentially much larger class of projective Finsler metrizable canonical sprays, corresponding to Lie groups, coincides with the class of Riemann metrizable canonical sprays. Generalisation of thes…
The Schwarzian derivative on Finsler manifolds of constant curvature
本文研究了常曲率 Finsler 流形上的 Schwarzian 导数，得到了 Möbius 方程的可积性条件、正常数 Ricci 曲率下的刚性结果，并证明了 Schwarzian 导数能刻画 Bonnet-Mayer 型定理。
Canonical frames in contact 3-manifolds and applications
受Cartan结构方程启发，研究接触3-流形上的特殊全局标架，该标架对偶于Bryant定义的广义Finsler结构，并利用标架相关的标量函数K给出刚性结果，应用于Reeb轨道作用量的尖锐估计。
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