Locally Desarguesian spaces

In Riemannian spaces, locally Desarguesian spaces have constant curvature and are therefore locally symmetric. This does not hold for Finsler spaces, so that locally Desarguesian spaces represent a generalization other than the obvious one we studied previously of (certain) Riemannian symmetric spaces. In this paper we discuss them in detail; as an example of the results obtained we mention that a simply connected locally Desarguesian space without conjugate points is globally Desarguesian. Applications are then given to spaces which are locally symmetric in a wider sense. We also study (and in Minkowski spaces determine exactly) the properties of functions which measure the distance of a point from those on a line.     

On generalized symmetric Finsler spaces

In this paper, we study generalized symmetric Finsler spaces. We first study some existence theorems, then we consider their geometric properties and prove that any such space can be written as a coset space of a Lie group with an invariant Finsler metric. Finally we show that each generalized symmetric Finsler space is of finite order and those of even order reduce to symmetric Finsler spaces and hence are Berwaldian.     

A projectively symmetric Finsler space

Symmetric (Riemannian) spaces were introduced and developed by Cartan [1, 2] which led to the discovery of projectively symmetric (Riemannian) spaces by Soós [9]. Recently the theory of symmetric spaces has been extended to Finsler geometry by the present author [5]. The current paper deals with that class of Finsler spaces throughout which their projective curvature tensors possess vanishing covariant derivatives. Following Soós' terminology such spaces are calledprojectively symmetric Finsler spaces. Examples, conditions for a symmetric Finsler space to be projectively symmetric, reduction of various identities, and the discussion of a decomposed projectively symmetric Finsler space form the skeleton of the paper.     

关于射影平坦Finsler空间

本文研究了射影平坦Finsler空间的几何量及其几何性质。证明了射影平坦Finsler空间的Ricci曲率可完全由射影因子简洁地刻画出来。同时还证明了，在射影平坦Finsler空间中，平均Berwald曲率S=0意味着Ricci曲率Ric是二次齐次的。此外，给出了一个射影平坦Finsler空间成为常曲率空间或局部Minkowski空间的充分条件。     

某些芬斯拉空间的对称性质

<正> E.嘉当在他的黎曼几何教程中系统地讨论了对称的黎曼空间,并给出了充要条件的分析形式及一系列有趣的性质.本文在芬斯拉空间中引进了嘉当在黎曼几何中所定义的“对称”概念后(第一节),对这类芬斯拉空间的对称性质作了详尽的讨论.得到的结果如下:(一)在 F_n 的一区域Ω内,把任一向量关于0点(O∈Ω)作对称推移和沿经过0的极值曲线作平行推移(以后在不引起混淆的情祝下,简称为“向量经过平行推移及对称推移”),为使这时所得结果之差为三阶小量,充要条件是:挠率张量的共变导数在Ω中等于零.E.Cartan 对这种空间巳作了一些几何说明,而这里给了一个新的几何特征.我们称这样的芬斯拉空间为亚对称的,黎曼空间即口为其中最常见的一个.     

On the classification of weakly symmetric Finsler spaces

In this paper, we give the classification of some special types of weakly symmetric Finsler spaces. We first present a general principle to classify weakly symmetric Finsler spaces and also give a method to figure out the Berwald spaces among the class of weakly symmetric Finsler spaces. Then we give an explicit classification of weakly symmetric Finsler spaces with reductive isometric groups as well as the left invariant weakly symmetric Finsler metrics on nilpotent Lie groups of the Heisenberg type. As an application, we obtain a large number of high-dimensional examples of reversible Finsler spaces which are non-Berwaldian and with vanishing S-curvature, a kind of spaces which are sought after in an open problem of Z. Shen.     

Deviations from parallelism and equidistance in Finsler space

In the present paper the deviations from parallelism and equidistance are studied for Finsler spaces in Cartan’s sense and for Finsler space which is locally Minkowskian. By considering an orthogonal ennuple of hypersurfaces in F_n properties of associate curvature or angular spread vector have been studied.     

Four-Dimensional Osserman Symmetric Spaces

It is shown that locally symmetric four-dimensional Osserman spaces are flat, locally isometric to a rank-one symmetric space or locally isometric to certain rank-two symmetric spaces.     

标量曲率Finsler空间与Finsler度量的射影变换

本文研究了与一个Ricci平坦Finsler空间或一个常曲率Finsler空间射影相关的标量曲率Finsler空间．我们给出了这种标量曲率Finsler空间成为常曲率空间的充分必要条件.特别地，我们给出了射影平坦Finsler空间具有常曲率的条件．     

A Cartan–Hadamard Theorem for Banach–Finsler Manifolds

In this paper we study Banach–Finsler manifolds endowed with a spray which have seminegative curvature in the sense that the corresponding exponential function has a surjective expansive differential in every point. In this context we generalize the classical theorem of Cartan–Hadamard, saying that the exponential function is a covering map. We apply this to symmetric spaces and thus obtain criteria for Banach–Lie groups with an involution to have a polar decomposition. Typical examples of symmetric Finsler manifolds with seminegative curvature are bounded symmetric domains and symmetric cones endowed with their natural Finsler structure which in general is not Riemannian.     

Dually flat general spherically symmetric Finsler metrics

Dually flat Finsler metrics arise from information geometry which has attracted some geometers and statisticians. In this paper, we study dually flat general spherically symmetric Finsler metrics which are defined by a Euclidean metric and two related 1-forms. We give the equivalent conditions for those metrics to be locally dually flat. By solving the equivalent equations, a group of new locally dually flat Finsler metrics is constructed.     

Differentiable Distance Spaces

The distance function \({\varrho(p, q) ({\rm or} d(p, q))}\) of a distance space (general metric space) is not differentiable in general. We investigate such distance spaces over \({\mathbb{R}^n}\), whose distance functions are differentiable like in case of Finsler spaces. These spaces have several good properties, yet they are not Finsler spaces (which are special distance spaces). They are situated between general metric spaces (distance spaces) and Finsler spaces. We will investigate such curves of differentiable distance spaces, which possess the same properties as geodesics do in Finsler spaces. So these curves can be considered as forerunners of Finsler geodesics. They are in greater plenitude than Finsler geodesics, but they become geodesics in a Finsler space. We show some properties of these curves, as well as some relations between differentiable distance spaces and Finsler spaces. We arrive to these curves and to our results by using distance spheres, and using no variational calculus. We often apply direct geometric considerations.     

二维Finsler空间是共形平坦的若干判定条件

本文利用Finsler约度量函数与度量张量获得了二维Finsler空间是共形平坦的若干令新的充要条件．此外，还推导了在共形映射下，局部Minkowski空间、常曲率Finsler空间与零曲率Finsler空间保持不变的新的充要条件．     

Polyhedral Finsler spaces with locally unique geodesics

We study Finsler PL spaces, that is simplicial complexes glued out of simplices cut off from some normed spaces. We are interested in the class of Finsler PL spaces featuring local uniqueness of geodesics (for complexes made of Euclidean simplices, this property is equivalent to local CAT(0)). Though non-Euclidean normed spaces never satisfy CAT(0), it turns out that they share many common features. In particular, a globalization theorem holds: in a simply-connected Finsler PL space local uniqueness of geodesics implies the global one. However the situation is more delicate here: some basic convexity properties do not extend to the PL Finsler case.     

Hypersurfaces of semi-C-reducible and S4-like Finsler spaces

The notions of semi C-reducible and S4-like Finsler spaces have been introduced by Matsumoto and Shibata ([6]). The object of the present paper is to study some properties of the hypersurfaces immersed in semi-C-reducible and S4-like Finsler spaces. It has been proved that a hypersurface of semi-C-reducible Finsler space is a semi-C-reducible while the condition, under with a hypersurface of S4-like Finsler space will be a S-4like space, has been obtained. The condition under which a hypersurface of semi-C-reducible Landsberg space will be a Landsberg space has also been obtained. After using the so called “T-condition” (Matsumoto [5]) we have discussed the condition under which a hypersurface of a semi-C-reducible Finsler spaceF _n satisfying T-condition will also satisfy T-condition.     

Unit Tangent Sphere Bundles and Two-Point Homogeneous Spaces

We characterize two-point homogeneous spaces, locally symmetric spaces, C and B-spaces via properties of the standard contact metric structure of their unit tangent sphere bundle. Further, under various conditions on a Riemannian manifold, we show that its unit tangent sphere bundle is a (locally) homogeneous contact metric space if and only if the manifold itself is (locally) isometric to a two-point homogeneous space.     

On projectively flat Finsler spaces

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 field (Theorem 3). In the last section we present a characterization of the Finsler spaces which are projectively flat in a parameter-preserving manner (Theorem 4), and we show that these spaces over ${\mathbb {R}}^{n}$ are exactly the Minkowski spaces (Theorems 5 and 6).