A. Soleiman
The aim of the present paper is to investigate new types of recurrence in Finsler geometry, namely, hyper-generalized recurrence and generalized conharmonic recurrence. The properties of such recurrences and their relations to other Finsler recurrences are studied.
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In this paper we adopt the pullback approach to global Finsler geometry. We investigate horizontally recurrent Finsler connections. We prove that for each scalar ($π$)1-form $A$, there exists a unique horizontally recurrent Finsler connection whose $h$-recurrence form is $A$. This result generalizes the existence and uniqueness theorem of Cartan connection. We then study some properties of a special kind of horizontally recurrent Finsler connection, which we call special HRF-connection.
On Concircularly Recurrent Finsler Manifolds
引入并研究了两种特殊的Finsler空间，即$R^h$ -递归Finsler空间和一致递归Finsler空间.利用Cartan联络的第一曲率张量给出了这些空间的定义性质。以下三个结果构成了本论文的主要研究对象：1.一个共圆平坦的芬斯勒流形必然具有常曲率（定理A）; 2.每一个$R^h$ -递归Finsler流形都具有相同的递归形式共圆递归（定理B）; 3.每个水平可积的共圆递归Finsler流形都是具有相同递归形式的$R^h$ -递归的（定理C）.整个工作是在一个坐标自由的形式制定。
The pullback approach to global Finsler geometry is adopted. Some new types of special Finsler spaces are introduced and investigated, namely, Ricci, generalized Ricci, projectively recurrent and m-projectively recurrent Finsler spaces. The properties of these special Finsler spaces are studied and the relations between them are singled out.
Conformal change of special Finsler spaces
本文是前一篇论文[Tensor，N.美国，69（2008），155-178]。主要目的是建立最重要的特殊Finsler空间的共形变换的\emph{an intrinsic investigation}，即$C^{h}$ -常返的，$C^{v}$ -常返的，$C^{0}$ -常返的，$C_{2}$ -类的，拟$C$-可约的，$C$ -可约的，Berwald空间，$S^{v}$ -常返的，$P^*$-Finsler流形，$R_{3}$ -类的，$P$ -对称的，$p$ -纯量曲率的Finsler流形和$s$ - $ps$ -曲率的Finsler流形。得到了这类特殊Finsler流形在共形变换下不变的充要条件。此外，还给出了Chern联络和Hashiguchi联络的共形变换及其曲率张量。
Recently, we have studied the Finsler space with h-Matsumoto change and found Cartan connection for the transformed space [2]. In this paper, we have discussed certain geometrical properties of the hypersurface of a Finsler space for the h-Matsumoto change.
Recently we have obtained the Cartan connection for the Finsler space whose metric is given by an exponential change with an h-vector. In this paper, we discuss certain geometric properties of a Finslerian hyperspace subjected to an h-exponential change of metric.
In this paper, we study the relationship between isoparametric hypersurfaces and hypersurfaces with constant principal curvatures in Finsler spaces. We give some examples of isoparametric hypersurfaces with (non)constant principal curvatures on Randers manifolds with nonconstant flag curvatures. Furthermore, we construct an example of a conformally flat Randers manifold which admits a family of nonisoparametric hyperplanes with constant principal curvatures.
Special Finsler spaces admitting a semi-concurrent vector field
本文研究芬斯勒流形上的半共线向量场，证明四类特殊芬斯勒空间在允许该向量场时等价于黎曼空间，并给出满足$C$-共形条件时成为黎曼空间的充要条件。
In this paper, we study isoparametric hypersurfaces in Finsler space forms by investigating focal points, tubes and parallel hypersurfaces of submanifolds. We prove that the focal submanifolds of isoparametric hypersurfaces are anisotropic-minimal and obtain the Cartan-type formula in a Finsler space form with vanishing reversible torsion, from which we give some classifications on the number of distinct principal curvatures or their multiplicities.
A Global Approach to the Theory of Special Finsler Manifolds
本文的目的是提供一个特殊的Finsler流形理论的整体介绍。我们在全球范围内引进和调查（或本质上，从局部坐标自由）许多最重要和最常用的特殊芬斯勒流形：局部闵可夫斯基，Berwald，Landesberg，一般Landesberg，$P$ -可约，$C$ -可约，半- $C$-可约，准- $C$ -可约，$P^{*}$ -Finsler，$C^{h}$-递归，$C^{v}$ -递归，$C^{0}$ -递归，$S^{v}$ -递归，$S^{v}$ -二阶递归，$C_{2}$ -类，$S_{3}$ -类，$S_{4}$ -类，$P_{2}$ -类，$R_{3}$ -类，$P$ -对称，$h$ -各向同性，标量曲率，常曲率，$p$ -标量曲率，$s$-$ps$-曲率.给出了这些特殊Finsler流形的整体定义。我们发现了所考虑的特殊Finsler流形的不同类型之间的各种关系，并在整体上证明了许多文献中已知的局部结果，得到了一些新的结果。作为副产品，我们得到了挠率张量场和曲率张量场的一些有趣的恒等式和性质。虽然我们的调查是完全全球性的，我们提供;比较的原因，附录提出了一个本地对应的我们的全球性的方法和本地定义的特殊芬斯勒空间考虑。
In this paper, we introduce isoparametric functions and isoparametric hypersurfaces in conic Finsler spaces. We find that in a conic Minkowski space, besides the conic Minkowski hyperplanes, conic Minkowski hyperspheres and conic Minkowski cylinders, which are all isoparametric hypersurfaces, there are probably other isoparametric hypersurfaces, such as helicoids. Moreover, we give a complete classification of isoparametric hypersurfaces in kropina spaces with constant flag curvature.
Hypersurface of a Finsler space subjected to a Kropina change with an h-vector
h-向量的概念是由H. 1980年的泉。最近我们得到了Finsler空间的Cartan联络，其度量是由具有h-向量的Kropina变换给出的。1985年，M.松本研究芬斯勒超曲面理论。本文给出了具有h-向量的Kropina变换的Finslerian超曲面的若干几何性质
Hypersurfaces with Constant Mean Curvatures on Finsler manifolds
该研究在Finsler流形框架下，通过保体积变分明确常平均曲率超曲面的几何意义，建立同伦导航下子流形主曲率的对应关系，还推导了Heintze-Karcher型不等式并证明特殊Finsler空间中的Alexandrov型定理。
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…
Theory of hypersurfaces of a Finsler space with generalized square metric
本文研究了广义平方度量 F(x,y)=(α(x,y)+β(x,y))^(n+1)/(α^n (x,y)) 在 Finsler 超曲面中的应用，并确立了 Finsler 流形三类超平面的分类与存在性。
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。
Isoparametric functions on Finsler space forms
本文在Finsler空间形式$(N(c),F)$上证明在一定条件下超正则函数是等参函数，推广了Riemann情形的定理B；并在前向Finsler空间形式$(\overrightarrow{N}(c),F)$上讨论脐超曲面与等参超曲面的关系，利用距离函数构造全局和局部等参函数。
The Funk-Finsler Structure in the Constant Curvature Spaces
本文研究常曲率空间中的Funk-Finsler结构，通过计算多种曲率揭示其几何性质，给出了不同常曲率空间下S-曲率和旗曲率的界值与欧氏情形的恒值结论。
We consider a natural mechanical system on a Finsler manifold and study its \emph{curvature} using the intrinsic Jacobi equations (called \emph{Jacobi curves}) along the extremals of the least action of the system. The curvature for such a system is expressed in terms of the Riemann curvature and the Chern curvature (involving the gradient of the potential) of the Finsler manifold and the Hessian of the potential w.r.t. a Riemannian metric induced from the Finslerian metric. As an application, we give sufficient conditions for the Hamiltonian flows of the least action to be hyperbolic and show…
Isoparametric Hypersurfaces in Minkowski Spaces
本文在Finsler流形中引入等参函数和等参超曲面，并给出了超正规函数是等参函数的充要条件。然后证明了具有$BH$ -体积的Minkowski空间中的超平面、Minkowski超球和$F^*$ -Minkowski柱（分别为. $HT$ -体积）形式都是分别具有一个和两个不同的常数主曲率的等参超曲面。此外，我们还给出了Randers-Minkowski空间中等参超曲面的一个完全分类，并构造了一个反例，证明了Wang在\cite{WQ}中的定理B在Finsler几何中不成立.
We show how some of well-known recurrent operators such as recurrent curvature operator, recurrent Ricci operator, recurrent Jacobi operator, recurrent shape and Weyl operators have the significant role for biharmonic hypersurfaces to be minimal in the Euclidean space.
Some remarks on Finsler manifolds with constant flag curvature
本文是对具有常正旗曲率的Finsler流形几何的四个松散相关的评论的阐述。<p>第一个注记是在这样一个流形的测地线空间上存在一个典型的Kahler结构。<p>第二点是，有一种自然的方法可以从复射影n-空间中适当一般位置的超曲面中构造一个（不一定是完备的）具有常正旗曲率的芬斯勒n-流形。<p>第三个注记是2-球面上常曲率的Finsler度量可用其测地线空间上的黎曼度量和1-形式来描述。特别地，这允许使用2-球面上正高斯曲率的任何（黎曼）Zoll度量来构造2-球面上常正曲率的全局Finsler度量。<p>第四点是关于维数n+1>2中具有常正旗曲率的（局部）Finsler度量空间的一般性。证明了这样的度量依赖于n（n+1）个n+1元的任意函数，并且这样的度量自然地对应于2n-流形上的某些无挠S^1 x GL（n，R）-结构.作为一个副产品，它被发现，这些群体确实出现的holonomy的挠自由仿射联络在2n维，迄今未被怀疑的现象。
Real Hypersurfaces in complex two-plane Grassmannians with recurrent Ricci tensor
本文在复二平面Grassmannian算子$G_2({\mathbb C}^{m+2})$中引入了广义Tanaka-Webster Reeb递归Ricci张量的新概念.其次，在这样的条件下，我们给出了实超曲面$M$在$G_2({\mathbb C}^{m+2})$中的不存在性。
What is wrong with the Hausdorff measure in Finsler spaces
我们在三维空间中构造了一类Finsler度量，使得它们的测地线都是直线，但并不是所有的平面都是Hausdorff面积泛函的极值。这表明，如果Hausdorff测度被用作Finsler空间上的体积概念，则全测地子流形不一定是极小的，从而填补了S.伊万诺夫不举行，积分几何公式不存在。另一方面，利用Holmes-Thompson体积定义，证明了Finsler空间的一个广义Crofton公式，并给出了Finsler空间的全测地超曲面是极小的简单证明.
Generalized Berwald manifolds are Finsler manifolds admitting linear connections such that the parallel transports preserve the Finslerian length of tangent vectors. By the fundamental result of the theory \cite{V5} such a linear connection must be metrical with respect to the averaged Riemannian metric given by integration of the Riemann-Finsler metric on the indicatrix hypersurfaces. Therefore the linear connection is uniquely determined by its torsion tensor. If the torsion is zero then we have a classical Berwald manifolds. Otherwise the torsion is a strange data we need to express in term…
On the interplay between Lorentzian Causality and Finsler metrics of Randers type
利用标准定态时空在$M=\R\times S$上的共形结构（因果性）与在$S$上的Randers度规之间的对应关系，得到了Lorentz几何和Finsler几何中的一些结果.特别是，对于平稳时空，我们给出了一个简单的特征，当它们是因果连续或全局双曲（包括在后一种情况下，当$S$是柯西超曲面），在相关的Randers度量。柯西发展的可计算性的后果也来自。因果关系表明，黎曼几何的许多结果（通过最小化测地线的测地线连通性，Bonnet-Myers，Synge定理）中的完备性的作用在芬斯勒几何中是由对称闭球的紧性发挥的。在此条件下，我们证明了对任意的Randers度量，存在另一个具有相同的预测地线且测地线完备的Randers度量。更重要的是，柯西视界在时空中的可微性的结果产生的后果的可微性的兰德斯距离的子集，反之亦然。
Generalized Berwald manifolds are Finsler manifolds admitting linear connections such that the parallel transports preserve the Finslerian length of tangent vectors (compatibi\-li\-ty condition). By the fundamental result of the theory \cite{V5} such a linear connection must be metrical with respect to the averaged Riemannian metric given by integration of the Riemann-Finsler metric on the indicatrix hypersurfaces. Therefore the linear connection (preserving the Finslerian length of tangent vectors) is uniquely determined by its torsion. If the torsion is zero then we have a classical Berwald …
Curvatures of spheres in Hilbert geometry
证明了Hilbert几何中超球面的法曲率、圆的Rund曲率和Finsler曲率随着半径趋于无穷大而趋于1
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。
A new Laplace operator in Finsler geometry and periodic orbits of Anosov flows
在本文的第一部分中，我们给出了Finsler度量的Laplace算子的一个新的定义，它是二阶方向导数关于角测度的平均。该算子是椭圆形的，对称的霍姆斯-汤普森体积，并符合通常的拉普拉斯-贝尔特拉米算子时，芬斯勒度量是黎曼。我们计算一些Katok-Ziller度量的显式谱数据。当芬斯勒度量是负弯曲的，我们表明，由于安科纳的结果，马丁边界是Hölder同胚的视觉边界。这使我们能够推出调和测度的存在性和一些遍历性质。本文的第二部分研究了三维流形中的Anosov流，其中叶空间同胚于$\mathbb{R}$。当流形是双曲流形时，Thurston证明了（不）稳定的叶理诱导了一个调节流。我们利用这第二流来研究Anosov流的周期轨道的合痕类和嵌入圆柱的存在性。
The present paper deals with the proper existence of a generalized class of recurrent manifolds, namely, hyper-generalized recurrent manifolds. We have established the proper existence of various generalized notions of recurrent manifolds. For this purpose we have presented a metric and computed its curvature properties and finally we have obtained the existence of a new class of semi-Riemannian manifolds which are non-recurrent but hyper-generalized recurrent, Ricci recurrent, conharmonically recurrent, manifold of recurrent curvature 2-forms and semisymmetric; not weakly symmetric but weakly…
Ideal Poisson--Voronoi tessellations for Finsler metrics on symmetric spaces
本文研究高阶对称空间上芬斯勒度量的理想泊松-沃罗诺伊镶嵌，证明多面体芬斯勒度量对应的镶嵌可呈现前人发现的两种行为，并引入具有良好性质的理想塞尔伯格-泊松-沃罗诺伊镶嵌。
Generalized Finsler geometry and the anisotropic tearing of skin
提出基于广义Finsler几何的连续介质理论，用纤维丛、内部状态向量和相场耦合超弹性描述皮肤微结构演化与各向异性撕裂。
On super generalized recurrent manifolds
为了推广常返流形的概念，在文献中有各种各样的类常返条件。本文提出了一个类递归结构，即\textit{super generalized recurrent manifold}，它推广了超广义递归流形和弱广义递归流形.本文主要研究超广义回归流形的几何性质。最后，为了保证这种结构的存在性，我们通过适当的度量给出了一个适当的例子。
The optimal transport problem is studied in the context of Lorentz-Finsler geometry. For globally hyperbolic Lorentz-Finsler spacetimes the first Kantorovich problem and the Monge problem are solved. Further the intermediate regularity of the transport paths is studied. These results generalize parts of Bertrand & Puel and Brenier et al.
On Warped Product Super Generalized Recurrent Manifolds
本文的目的是得到具有一类特殊的类递归结构（称为超广义递归结构）的翘积半黎曼流形的特征。作为这个结果的推论，我们还得到了翘积流形满足弱广义递归流形、超广义递归流形等其他类递归结构的充要条件.最后，作为主要结果的支持，我们给出了翘积超广义递归流形的一个例子.
On the homology length spectrum of surfaces
在具有Finsler度量的曲面上，研究了在同一同调类中，使长度最小的所有测地线多重曲线中，长度小于$L$的闭测地线的数目的渐近增长性.一类重要的表面是我们感兴趣的双曲曲面。
$β$-Uniform Convexity and Divisible Domains
本文证明严格凸可除域上的自然芬斯勒度量具有$\eta$-一致凸性，常数$\eta$与边界正则性相关，并借助AI辅助证明希尔伯特度量在适当假设下也满足该性质。
Let $Γ'<Γ$ be two discrete groups acting properly by isometries on a Gromov-hyperbolic space $X$. We prove that their critical exponents coincide if and only if $Γ'$ is co-amenable in $Γ$, under the assumption that the action of $Γ$ on $X$ is strongly positively recurrent, i.e. has a growth gap at infinity. This generalizes all previously known results on this question, which required either $X$ to be the real hyperbolic space and $Γ$ geometrically finite, or $X$ Gromov hyperbolic and $Γ$ cocompact. This result is optimal: we provide several counterexamples when the action is not strongly posi…
V-representation for normality equations in geometry of generalized Legendre transformation
正规方程描述了允许超曲面法向移动的牛顿动力系统。这些方程最早是在欧几里得几何中推导出来的。然后很快他们被重新推导出黎曼和芬斯勒几何。最近，我发现正规方程可以从经典和/或广义勒让德变换给出的几何中导出。然而，在这种情况下，它们似乎是用p-表示来写的，即。e.动量向量和它的分量。本文的目的是将正规方程转换回v-表示，这对于牛顿动力系统来说更自然。
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