Julius Lang
We show that in dimension 2 every Finsler metric with at least 3-dimensional Lie algebra of projective vector fields is locally projectively equivalent to a Randers metric. We give a short list of such Finsler metrics which is complete up to coordinate change and projective equivalence.
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
We study the topology of complete Finsler manifolds admitting convex functions
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。
The class of spherically symmetric Finsler metrics is studied and locally dually flat and projectively flat spherically symmetric Finsler metrics is classified.
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.
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。
We use two non-Riemannian curvature tensors, the $χ$-curvature and the mean Berwald curvature to characterise a class of Finsler metrics admitting first integrals.
A projective parameter of a geodesic on a Finsler space is defined to be solution of a certain ODE. Using projective parameter and Funk metric, one can construct a projectively invariant intrinsic pseudo-distance on a Finsler space. In the present work, solutions of the projective parameter's ODE are characterized with respect to the sign of parallel Ricci tensor of a Finsler space. It is shown that the pseudo-distance is trivial on complete Finsler spaces of positive semi-definite Ricci tensor and it is a distance on Finsler spaces of parallel negative definite Ricci tensor. These results gen…
We define a Weyl-type curvature tensor of $(1,2)$-type to provide a characterization for Finsler metrics of constant flag curvature. This Weyl-type curvature tensor is projective invariant only to projective factors that are Hamel functions. Based on this aspect we construct new families of projectively related Finsler metrics that have constant flag curvature.
In this paper we introduce in study the projectively related complex Finsler metrics. We prove the complex versions of the Rapcsák's theorem and characterize the weakly Kähler and generalized Berwald projectively related complex Finsler metrics. The complex version of Hilbert's Fourth Problem is also pointed out. As an application, the projectiveness of a complex Randers metric is described.
We investigate projective spherically symmetric Finsler metrics with constant flag curvature in $R^n$ and give the complete classification theorems. Furthermore, a new class of Finsler metrics with two parameters on n-dimensional disk are found to have constant negative flag curvature.
We study the cylindrical symmetric Finsler metrics. We obtain the system of differential equations of such metrics which are projectively flat. We give a family of solutions of this system. Examples are included.
We show that the infinite-dimensional space of Zoll Finsler metrics on the projective plane strongly deformation retracts to the canonical round metric. In particular, this space of Zoll Finsler metrics is connected. Moreover, the strong deformation retraction arises from a deformation of the geodesic flow of every Zoll Finsler projective plane to the geodesic flow of the round metric through a family of smooth free circle actions induced by the curvature flow of the canonical round projective plane. This construction provides a description of the geodesics of the Zoll Finsler metrics along th…
In this paper, we study a class of Finsler metrics composed by a Riemann metric $α=\sqrt{a_{ij}(x)y^i y^j}$ and a $1$-form $β=b_i(x)y^i$ called general ($α$, $β$)-metrics. We classify those projectively flat when $α$ is projectively flat. By solving the corresponding nonlinear PDEs, the metrics in this class are totally determined. Then a new group of projectively flat Finsler metrics is found.
We define a Weyl-type curvature tensor that provides a characterisation for Finsler metrics of constant flag curvature. When the Finsler metric reduces to a Riemannian metric, the Weyl-type curvature tensor reduces to the classic projective Weyl tensor. In the general case, the Weyl-type curvature tensor differs from the Weyl projective curvature, it is not a projective invariant, and hence Beltrami Theorem does not work in Finsler geometry. We provide the relation between the Weyl-type curvature tensors of two projectively related Finsler metrics. Using this formula we show that a projective …
In this paper we prove that the holonomy group of a simply connected locally projectively flat Finsler manifold of constant curvature is a finite dimensional Lie group if and only if it is flat or it is Riemannian.
In this paper, we find the necessary and sufficient conditions under which two classes of (q, a, b)-metrics are projectively related to a Kropina metric.
In this paper the projective curvature invariants of a complex Finsler space are obtained. By means of these invariants the notion of complex Douglas space is then defined. A special approach is devoted to obtain the equivalence conditions that a complex Finsler space should be Douglas. It is shown that any weakly Kähler Douglas space is a complex Berwald space. A projective curvature invariant of Weyl type characterizes the complex Berwald spaces. They must be either purely Hermitian of constant holomorphic curvature or non purely Hermitian of vanish holomorphic curvature. The locally project…
We proof that on a surface of negative Euler characteristic, two real-analytic Finsler metrics have the same unparametrized oriented geodesics, if and only if they differ by a scaling constant and addition of a closed 1-form.
Recently, we developed a method for the study of holonomy properties of non-Riemannian Finsler manifolds and obtained that the holonomy group can not be a compact Lie group, if the Finsler manifold of dimension $> 2$ has non-zero constant flag curvature. The purpose of this paper is to move further, exploring the holonomy properties of projectively flat Finsler manifolds of non-zero constant flag curvature. We prove in particular that projectively flat Randers and Bryant-Shen manifolds of non-zero constant flag curvature have infinite dimensional holonomy group.
In this paper, we study locally projectively flat Finsler metrics with constant flag curvature ${\bf K}$. We prove those are totally determined by their behaviors at the origin by solving some nonlinear PDEs. The classifications when ${\bf K}=0$, ${\bf K}=-1$ and ${\bf K} =1$ are given respectively in an algebraic way. Further, we construct a new projectively flat Finsler metric with flag curvature ${\bf K}=1$ determined by a Minkowskian norm with double square roots at the origin. As an application of our main theorems, we give the classification of locally projectively flat spherical symmetr…
In this paper, we use the technique of Finslerian submersion to deduce a flag curvature formula for homogeneous Finsler spaces. Based on this formula, we give a complete classification of even-dimensional smooth coset spaces $G/H$ admitting $G$-invariant Finsler metrics with positive flag curvature. It turns out that the classification list coincides with that of the even dimensional homogeneous Riemannian manifolds with positive sectional curvature obtained by N.R. Wallach. We also find out all the coset spaces admitting invariant non-Riemannian Finsler metrics with positive flag curvature.
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…
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.
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 …
In this paper, it is proved that any conformal vector field is homothetic on a locally projectively flat $(α,β)$-space of non-Randers type in dimension $n\ge 3$, and the local solutions of such a vector field are determined. While on a locally projectively flat Randers space, examples showthat the conformal vector fields are not necessarily homothetic.
On Locally Dually and Projectively Flat Almost Rational Finsler Metrics
研究几乎有理Finsler度量的局部对偶平坦和射影平坦的充要条件，并刻画同时满足两者的度量，提出局部Minkowski性猜想。
On projective symmetries on Finsler Spaces
本文证明了Akbar-Zadeh的两种Einstein-Finsler空间定义在$I$-不变射影向量场积分曲线上等价，揭示了闭Finsler空间中射影对称性退化为Killing向量场的条件，并完成了紧致各向同性平均Landsberg流形的分类。
Singular Finsler metrics, such as Kropina metrics and $m$-Kropina metrics, have a lot of applications in the real world. In this paper, we classify a class of singular $(α,β)$-metrics which are locally projectively flat with constant flag curvature in dimension $n= 2$ and $n \ge 3$ respectively. Further, we determine the local structure of $m$-Kropina metrics and particularly Kropina metrics which are projectively flat with constant flag curvature and prove that such metrics must be locally Minkowskian but are not necessarily flat-parallel.
We consider the projective Finsler metrizability problem: under what conditions the solutions of a given system of second-order ordinary differential equations (SODE) coincide with the geodesics of a Finsler metric, as oriented curves. SODEs with isotropic curvature have already been thoroughly studied in the literature and have proved to be projective Finsler metrizable. In this paper, we investigate the non-isotropic case and obtain new results by examining the integrability of the Rapcsák system extended with curvature conditions. We consider the $n$-dimensional generic case, where the eige…
An $(α,β)$-manifold $(M,F)$ is a Finsler manifold with the Finsler metric $F$ being defined by a Riemannian metric $α$ and $1$-form $β$ on the manifold $M$. In this paper, we classify $n$-dimensional $(α,β)$-manifolds (non-Randers type) which are positively complete and locally projectively flat. We show that the non-trivial class is that $M$ is homeomorphic to the $n$-sphere $S^n$ and $(S^n,F)$ is projectively related to a standard spherical Riemannian manifold, and then we obtain some special geometric properties on the geodesics and scalar flag curvature of $F$ on $S^n$, especially when $F$…
In this paper, we study a class of two-dimensional Finsler metrics defined by a Riemannian metric $α$ and a 1-form $β$. We characterize those metrics which are Douglasian or locally projectively flat by some equations. In particular, it shows that the known fact that $β$ is always closed for those metrics in higher dimensions is no longer true in two dimensional case. Further, we determine the local structures of two-dimensional $(α,β)$-metrics which are Douglassian, and some families of examples are given for projectively flat classes with $β$ being not closed.
Singular Finsler metrics, such as Kropina metrics and $m$-Kropina metrics, have a lot of applications in the real world. In this paper, we study a class of singular Finsler metrics defined by a Riemann metric $α$ and 1-form $β$ and characterize those which are respectively Douglasian and locally projectively flat in dimension $n\ge 3$ by some equations. Our study shows that the main class induced is an $m$-Kropina metric plus a linear part on $β$. For this class with $m\ne -1$, the local structure of projectively flat case is determined, and it is proved that a Douglas $m$-Kropina metric must …
We give a new and complete proof of the following theorem, discovered by Detlef Laugwitz: (forward) complete and connected finite dimensional Finsler manifolds admitting a proper homothety are Minkowski vector spaces. More precisely, we show that under these hypotheses the Finsler manifold is isometric to the tangent Minkowski vector space of the fixed point of the homothety via the exponential map of the canonical spray of the Finsler manifold.
In this paper, we give the general form of spherically symmetric Finsler metrics in $R^n$ and surprisedly find that many well-known Finsler metrics belong to this class. Then we explicitly express projective metrics of this type. The necessary and sufficient conditions that projective Finsler metrics with spherical symmetry have constant flag curvature are also obtained.
In the present paper, we find the conditions to characterize projective change between two $(α, β)$-metrics, F = $α+ εβ+ k\frac{β^2}α$ ($ε$ and k $\neq$ 0 are constants) and a Matsumoto metric $\bar{F}=\frac{\barα^2}{\barα -\barβ}$ on a manifold with dimension $n \geq 3$ where $α$ and $\barα$ are two Riemannian metrics, $β$ and $\barβ$ are two non-zero 1-forms. Moreover, we study such projective changes when F has some special curvature properties.
New classes of Finsler metrics: The birth of new projective invatiant
本研究在芬斯勒几何中提出新的射影不变量，通过构建W-GDW方程将GDW度量推广为更广泛的W-GDW度量类，含两个子类，阐明了这类度量的性质并辅以示例。
In this paper, we consider Randers change of some special $ (α, β)- $ metrics. First we find the fundamental metric tensor and Cartan tensor of these Randers changed $ (α, β)- $metrics. Next, we establish a general formula for inverse of fundamental metric tensors of these metrics. Finally, we find the necessary and sufficient conditions under which the Randers change of these $ (α, β)- $ metrics are projectively and locally dually flat.
In 2001, Zhongmin Shen asked if it is possible for two projectively related Finsler metrics to have the same Riemann curvature tensor, [14, page 184]. In this paper, we provide an answer to this question, within the class of Finsler metrics of scalar flag curvature. In Theorem 3.1, we show that the answer is negative, for non-vanishing scalar flag curvature. The answer is known to be positive when the scalar flag curvature vanishes, [12, 14] and this positive answer is related to the existence of many solutions to Hilbert's Fourth Problem. As a generalisation of this problem, we can ask if it …
Singular Finsler metrics, such as Kropina metrics and $m$-Kropina metrics, have a lot of applications in the real world. In this paper, we study a class of two-dimensional singular Finsler metrics defined by a Riemann metric $α$ and 1-form $β$, and we characterize those which are Douglasian or locally projectively flat by some equations. It shows that the main class induced is an $m$-Kropina metric plus a linear part on $β$. For this class, the local structure of Douglasian or (in part) projectively flat case is determined, and in particular we show that a Kropina metric is always Douglasian a…
In this paper, we use the flag curvature formula for homogeneous Finsler spaces in our previous work to classify odd dimensional smooth coset spaces admitting positively curved reversible homogeneous Finsler metrics. We will show that the most features of L. Bérard-Bergery's classification results for odd dimensional positively curved Riemannian homogeneous spaces can be generalized to reversible Finsler spaces.
In this paper, we find a condition under which a Finsler space with Kropina change of mth-root metric is projectively related to a mth-root metric and also we find a condition under which this Kropina transformed mth-root metric is locally dually flat. Moreover we find the condition for its Projective flatness.
In this paper, we study normal homogeneous Finsler spaces. We first define the notion of a normal homogeneous Finsler space, using the method of isometric submersion of Finsler metrics. Then we study the geometric properties. In particular, we establish a technique to reduce the classification of normal homogeneous Finsler spaces of positive flag curvature to an algebraic problem. The main result of this paper is a classification of positively curved normal homogeneous Finsler spaces. It turns out that a coset space $G/H$ admits a positively curved normal homogeneous Finsler metric if and only…
Generalized Weakly-Weyl Finsler Metrics: A Generalized Approach to Sakaguchi's Theorem
本文提出广义弱Weyl Finsler度量，建立其与射影不变量的联系，推广Sakaguchi定理，并证明弱Weyl与W-二次球对称Finsler度量在$\mathbb{R}^n$中等价。
In this paper, we prove that a strongly convex complex Finsler metric $F$ on a domain $D\subset\mathbb{C}^n$ is projectively flat (resp. dually flat) if and only if $F$ comes from a strongly convex complex Minkowski metric.
In this paper, we consider projective deformation of the geodesic system of Finsler spaces by holonomy invariant functions: Starting by a Finsler spray $S$ and a holonomy invariant function $P$, we investigate the metrizability property of the projective deformation $\widetilde{S}=S-2λP C$. We prove that for any holonomy invariant nontrivial function $P$ and for almost every value $λ\in R$, such deformation is not Finsler metrizable. We identify the cases where such deformation can lead to a metrizable spray: in these cases, the holonomy invariant function is necessarily one of the principal c…
In the year 1984 Shibata investigated the theory of a change which is called a $ β$-change of a Finsler metric. On the other hand in 1985 a systematic study of geometry of hypersurfaces in Finsler spaces was given by Matsumoto. In the present paper is to devoted to the study of a condition for a Randers conformal change to be projective and find out when a totally geodesic hypersurface $ F^{n-1} $ remains to be a totally geodesic hypersurface $ F^{n-1} $ under the projective Randers conformal change. Further obatined the condition under which a Finslerian hypersurfaces given by the projective …
A solution of Hilberts fourth problem lead to integral equation of the type generalized cosine transform. The present paper considers the solution that integral equation by integral geometry methods and propose an inversion formula for reconstruction of Crofton measures from projective smooth Finsler metrics in R3.
In this paper, it is proved that a connected 3-dimensional Riemannian manifold or a closed connected semi-Riemannian manifold $M^n$($n>1$) admitting a projective vector field with a non-linearizable singularity is projectively flat.
The holonomy of spherically symmetric projective Finsler metrics of constant curvature
本文证明常曲率球对称射影 Finsler 度量的完整群极大，并在单连通情形下同构于 $Diff_o({\mathbb S^{n-1}})$，首次显式刻画了非 Berwald 的 n 维 Finsler 流形完整群。
The aim of the present paper is to study the properties of Riemannian manifolds equipped with a projective semi-symmetric connection.
The projective Finsler metrizability problem deals with the question whether a projective-equivalence class of sprays is the geodesic class of a (locally or globally defined) Finsler function. In this paper we use Hilbert-type forms to state a number of different ways of specifying necessary and sufficient conditions for this to be the case, and we show that they are equivalent. We also address several related issues of interest including path spaces, Jacobi fields, totally-geodesic submanifolds of a spray space, and the equivalence of path geometries and projective-equivalence classes of spra…
In this paper, we studied a Finsler space whose metric is given by an h-exponential change and obtain the Cartan connection coefficients for the change. We also find the necessary and sufficient condition for an h-exponential change of Finsler metric to be projective.
It is the Hilbert's Fourth Problem to characterize the (not-necessarily-reversible) distance functions on a bounded convex domain in R^n such that straight lines are shortest paths. Distance functions induced by a Finsler metric are regarded as smooth ones. Finsler metrics with straight geodesics said to be projective. It is known that the flag curvature of any projective Finsler metric is a scalar function of tangent vectors (the flag curvature must be a constant if it is Riemannian). Thus it is a natural problem to study those of constant curvature. In this paper, we study the Hilbert Fourth…
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…
For non-reversible Finsler metrics of positive flag curvature on spheres and projective spaces we present results about the number and the length of closed geodesics and about their stability properties.
In Finsler geometry, there are infinitely many models of constant curvature. The Funk metrics, the Hilbert-Klein metrics and the Bryant metrics are projectively flat with non-zero constant curvature. A recent example constructed by the author is projectively flat with zero curvature. In this paper, we introduce a technique to construct non-projectively flat Finsler metrics with zero curvature in each dimension. The technique can be used to construct many non-projectively flat Finsler metrics of constant curvature.
The equation determining whether a projective structure admits a connection in its given projective class that has skew-symmetric Ricci tensor is an overdetermined system of semi-linear partial differential equations which we call the projective Einstein-Weyl (pEW) equation. In 2-dimensions, we give local obstructions for projective surfaces to admit such a connection in its projective class. The obstructions are the resultants of polynomial equations that have to be satisfied for there to admit any pEW solution.
In the current paper, first we give the correct version of the formula for mean Berwald curvature of a spherically symmetric Finsler metric given in paper \cite{YCheWSon2015}. Further, we establish differential equations characterizing projectively as well as dually flat spherically symmetric Finsler metrics. Finally, we obtain a rigidity result on spherically symmetric Finsler metrics with isotropic $E$-Curvature.
In this work we show that for the geodesic spray $S$ of a Finsler function $F$ the most natural projective deformation $\widetilde{S}=S -2 λF\mathbb C$ leads to a non-Finsler metrizable spray, for almost every value of $λ\in \mathbb R$. This result shows how rigid is the metrizablility property with respect to certain reparameterizations of the geodesics. As a consequence we obtain that the projective class of an arbitrary spray contains infinitely many sprays that are not Finsler metrizable.
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。