Georgeta Cretu
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.
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
The class of spherically symmetric Finsler metrics is studied and locally dually flat and projectively flat spherically symmetric Finsler metrics is classified.
In this work an intrinsic projectively invariant distance is used to establish a new approach to the study of projective geometry in Finsler space. It is shown that the projectively invariant distance previously defined is a constant multiple of the Finsler distance in certain case. As a consequence, two projectively related complete Einstein Finsler spaces with constant negative scalar curvature are homothetic. Evidently, this will be true for Finsler spaces of constant flag curvature as well.
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.
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.
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.
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 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.
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, 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.
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…
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.
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.
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 …
First we present a short overview of the long history of projectively flat Finsler spaces. We give a simple and quite elementary proof of the already known condition for the projective flatness, and we give a criterion for the projective flatness of a special Lagrange space (Theorem 1). After this we obtain a second-order PDE system, whose solvability is necessary and sufficient for a Finsler space to be projectively flat (Theorem 2). We also derive a condition in order that an infinitesimal transformation takes geodesics of a Finsler space into geodesics. This yields a Killing type vector fie…
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.
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。
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 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.
Let $M$ be a differentiable manifold and $TM$ be its tangent bundle. A $C^0$-Finsler structure on $M$ is a continuous function $F: TM \rightarrow \mathbb R$ such that its restriction to each tangent space is a norm. In this work we present a large family of projectively equivalent $C^0$-Finsler manifolds $(\hat M \cong \mathbb R^2,\hat F)$. Their structures $\hat F$ don't have partial derivatives and they aren't invariant by any transformation group of $\hat M$. For every $p,q \in (\hat M,\hat F)$, we determine the unique minimizing path connecting $p$ and $q$. They are line segments parallel …
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.
In this paper we study the flag curvature of a particular class of Finsler metrics called general $(α,β)$-metrics, which are defined by a Riemannian metric $α$ and a $1$-form $β$. The classification of such metrics with constant flag curvature are completely determined under some suitable conditions, which make them be locally projectively flat. As a result, we construct some new projectively flat Finsler metrics with flag curvature $1$, $0$ and $-1$, all of which are of singularity at some directions.
In this paper, we investigate the holonomy structure of the most accessible and demonstrative 2-dimensional Finsler surfaces, the Randers surfaces. Randers metrics can be considered as the solutions of the Zermelo navigation problem. We give the classification of the holonomy groups of locally projectively flat Randers two-manifolds of constant curvature. In particular, we prove that the holonomy group of a simply connected non-Riemannian projectively flat Finsler two-manifold of constant non-zero flag curvature is maximal and isomorphic to the orientation preserving diffeomorphism group of th…
I classify the Finsler structures on the 2-sphere that have constant Finsler-Gauss curvature and whose geodesics are the great circles. Modulo diffeomorphism, there is a 2-parameter family of such Finsler structures, only one of which is homogeneous or symmetric, namely the Riemannian one. I discuss the history of the problem and its relation with Hilbert's Fourth Problem and the calculus of variations.
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 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.
If the flag curvature of a Finsler manifold reduces to sectional curvature, then locally either the Finsler metric is Riemannian, or the flag curvature is isotropic.
We classify homogeneous reversible Finsler metrics with positive Flag curvature. We show that if G/H admits a G invariant reversible Finsler metric with positive Flag curvature, then up to a few low dimensional spaces, it also admits a G invariant Riemannian metric with positive sectional curvature. For the exceptions, we do not know if they admit homogeneous Finsler metrics with positive Flag curvature.
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 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.
A Riemannian metric is of constant curvature if and only if it is locally projectively flat. There are infinitely many locally projectively flat Finsler metrics of constant curvature, that are special solutions to the Hilbert's Fourth Problem. In this paper, we use the technique in the paper titled "Finsler metrics with K=0 and S=0" (math.DG/0109060) to construct infinitely many Finsler metrics on the 2-sphere with constant curvature K=1 and infinitely many Finsler metrics on the 2-disk with constant curvature K = -1. These metrics are not projectively flat. So far, the classification of Finsl…
In this paper, first, we give an explicit formula for the flag curvature of a homogeneous Finsler space with generalized $m$-Kropina metric. Then, we show that, under a mild condition, the two definitions of naturally reductive homogeneous Finsler space are equivalent for afore said metric. Finally, we study the flag curvature of naturally reductive homogeneous Finsler spaces with generalized $m$-Kropina metric.
Finsler manifolds with Positive Weighted Flag Curvature
该研究引入基于$T$-曲率修正的加权旗曲率概念，证明两类具正/非负加权旗曲率的Finsler流形分别微分同胚于欧氏空间与欧氏球，推进了Finsler几何的曲率与拓扑关联研究。
In this paper we characterize sprays that are metrizable by Finsler functions of constant flag curvature. By solving a particular case of the Finsler metrizability problem we provide the necessary and sufficient conditions that can be used to decide whether or not a given homogeneous system of second order ordinary differential equations represents the Euler-Lagrange equations of a Finsler function of constant flag curvature. The conditions we provide are tensorial equations on the Jacobi endomorphism. We identify the class of homogeneous SODE where the Finsler metrizability is equivalent with…
Spherically symmetric metrics form a rich and important class of metrics. Many well-known Finsler metrics of constant flag curvature can be locally expressed as a spherically symmetric metric on R^n. In this paper, we study spherically symmetric metrics with constant Ricci curvature and constant flag curvature.
In this paper, first we derive an explicit formula for the flag curvature of a homogeneous Finsler space with infinite series $(α, β)$-metric and exponential metric. Next, we deduce it for naturally reductive homogeneous Finsler space with the above mentioned metrics.
In this paper we prove that a Finsler metrics has constant flag curvature if and only if the curvature of the induced nonlinear connection satisfies an algebraic identity with respect to some arbitrary second rank tensors. Such algebraic identity appears as an obstruction to the formal integrability of some operators in Finsler geometry, [4,7]. This algebraic characterisation for Finsler metrics of constant flag curvature allows to provide yet another proof for the Finslerian version of Beltrami's Theorem, [2,3].
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.
We show that the results of Foulon (1997 an 2002) and Kim (2007) (independently, Deng and Hou (2007)) about the nonexistence of locally symmetric Finsler metrics of positive or negative flag curvature are in fact local.
In this paper we study spherically symmetric metrics on a symmetric space in $\mathbb{R}^n$ with scalar and constant flag curvature and we also obtain families of this type of metrics. Many explicit examples are provided for Douglas metrics with scalar and constant flag curvature. Furthermore, new examples of projectively flat Finsler metrics are given. We also provide a family of spherically symmetric Finsler metrics which are not of Douglas type.
Here, using the projectively invariant pseudo-distance and Schwarzian derivative, it is shown that every connected complete Finsler space of the constant negative Ricci scalar is reversible. In particular, every complete Randers metric of constant negative Ricci (or flag) curvature is Riemannian.
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 his book "Differential Geometry of Spray and Finsler spaces", page 177, Zhongmin Shen asks "wether or not there always exist non-trivial Funk functions on a spray space". In this note, we will prove that the answer is negative for the geodesic spray of a finslerian function of non-vanishing scalar flag curvature.
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.
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…
Finsler metrics of scalar flag curvature play an important role to show the complexity and richness of general Finsler metrics. In this paper, on an $n$-dimensional manifold $M$ we study the Finsler metric $F=F(x,y)$ of scalar flag curvature ${\bf K} = {\bf K}(x,y)$ and discover some equations ${\bf K}$ should be satisfied. As an application, we mainly study the metric $F$ of weakly isotropic flag curvature ${\bf K} = \frac{3 θ}{F} + σ$, where $θ=θ_i(x) y^i \neq 0$ is a $1$-form and $σ=σ(x)$ is a scalar function. We prove that in this case, $F$ must be a Randers metric when $dim(M) \geq 3$. Fu…
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…
In this paper, we introduce the flag-wise positively curved condition for Finsler spaces (the (FP) Condition), which means that in each tangent plane, we can find a flag pole in this plane such that the corresponding flag has positive flag curvature. Applying the Killing navigation technique, we find a list of compact coset spaces admitting non-negatively curved homogeneous Finsler metrics satisfying the (FP) Condition. Using a crucial technique we developed previously, we prove that most of these coset spaces cannot be endowed with positively curved homogeneous Finsler metrics. We also prove …
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 consider Kropina change of $m$-th root Finsler metrics. We find necessary and sufficient condition under which the Kropina change of an $m$-th root Finsler metric be locally dually flat. Then we prove that the Kropina change of an $m$-th root Finsler metric is locally projectively flat if and only if it is locally Minkowskian.
We prove that a homogeneous Finsler sphere with constant flag curvature $K\equiv1$ and a prime closed geodesic of length $2π$ must be Riemannian. This observation provides the evidence for the non-existence of homogeneous Bryant spheres. It also helps us propose an alternative approach proving that a geodesic orbit Finsler sphere with $K\equiv1$ must be Randers. Then we discuss the behavior of geodesics on a homogeneous Finsler sphere with $K\equiv1$. We prove that many geodesic properties for homogeneous Randers spheres with $K\equiv1$ can be generalized to the non-Randers case.
This paper has been withdrawn by the author due to a crucial sign error in equation 1. An isometry $ρ$ of a connected Finsler space $(M, F)$ is called bounded if the function $d(x, ρ(x))$ is bounded on $M$. It is called a Clifford-Wolf translation if the function $d(x, ρ(x))$ is constant on $M$. In this paper, we prove that on a complete connected simply connected Finsler space of non-positive flag curvature, an isometry is bounded if and only if it is a Clifford-Wolf translation. As an application, we prove that a homogeneous Finsler space of negative flag curvature admits a transitive solvab…
In this paper, we characterize locally dually flat generalized m-th root Finsler metrics. Then we find a condition under which a generalized m-th root metric is projectively related to a m-th root metric. Finally, we prove that if a generalized m-th root metric is conformal to a m-th root metric, then both of them reduce to Riemannian metrics.
In this paper, we consider a special class of singular Finsler metrics: $m$-Kropina metrics which are defined by a Riemannian metric and a $1$-form. We show that an $m$-Kropina metric ($m\ne -1$) of scalar flag curvature must be locally Minkowskian in dimension $n\ge 3$. We characterize by some PDEs a Kropina metric ($m=-1$) which is respectively of scalar flag curvature and locally projectively flat in dimension $n\ge 3$, and obtain some principles and approaches of constructing non-trivial examples of Kropina metrics of scalar flag curvature.
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 …
In this paper, we study the interaction between the geodesic orbit (g.o.~in short) property and certain flag curvature conditions. A Finsler manifold is called g.o.~if each constant speed geodesic is the orbit of a one-parameter subgroup. Besides the non-negatively curved condition, we also concern the (FP) condition for the flag curvature, i.e., in any flag we can find a flag pole, such that the flag curvature is positive. The main theorem we will prove is the following. If a g.o.~Finsler space $(M,F)$ has non-negative flag curvature and satisfies the (FP) condition, then $M$ must be compact.…
Very standard homogeneous Finsler manifolds with positive flag curvature
本文在紧连通单连通单Lie群的齐性流形上定义并推广了标准齐性Finsler度量，并分类了所有容许正旗曲率非常标准齐性Finsler度量的情形。
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。