Funk Metrics and R-Flat Sprays

Finsler Metrics with K=0 and S=0

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.

On Geodesics of Sprays and Projective Completeness

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…

On Sprays of Scalar Curvature and Metrizability

Every Finsler metric naturally induces a spray but not so for the converse. The notion for sprays of scalar (resp. isotropic) curvature has been known as a generalization for Finsler metrics of scalar (resp. isotropic) flag curvature. In this paper, a new notion, sprays of constant curvature, is introduced and especially it shows that a spray of isotropic curvature is not necessarily of constant curvature even in dimension $n\ge3$. Further, complete conditions are given for sprays of isotropic (resp. constant) curvature to be Finsler-metrizabile. As applications of such a result, the local str…

关于球对称喷射的研究

On Spherically Symmetric Sprays

本文研究了球对称喷射，即在正交变换下不变的喷射。通过建立其标准形式，证明其测地系数可表示为特定形式，并对具有各向同性曲率的投影平坦球对称喷射进行了完整分类，特别得到了零曲率喷射的显式形式。

Projective Metrizability and Formal Integrability

The projective metrizability problem can be formulated as follows: under what conditions the geodesics of a given spray coincide with the geodesics of some Finsler space, as oriented curves. In Theorem 3.8 we reformulate the projective metrizability problem for a spray in terms of a first-order partial differential operator $P_1$ and a set of algebraic conditions on semi-basic 1-forms. We discuss the formal integrability of $P_1$ using two sufficient conditions provided by Cartan-Kähler theorem. We prove in Theorem 4.2 that the symbol of $P_1$ is involutive and hence one of the two conditions …

关于Finsler流形上测地喷雾的F-协变系数

On the covariant coefficients of geodesic sprays on Finsler manifolds

本文引入Finsler度量测地喷雾的F-协变系数与S-标量概念，证明S-标量存在当且仅当度量对偶平坦，并研究射影平坦情形下的线性联络性质，进而提出H-Berwald与H-Landsberg空间并解决相关Landsberg问题。

关于Berwald-Weyl曲率

On the Berwald-Weyl Curvature

本文研究带体积形式的spray/Finsler度量的Berwald-Weyl曲率，推导其表达式并证明其为射影不变量，给出两类该曲率为零的情形，提出BWeyl-平坦的新概念。

Metrizable isotropic second-order differential equations and Hilbert's fourth problem

It is well known that a system of homogeneous second-order ordinary differential equations (spray) is necessarily isotropic in order to be metrizable by a Finsler function of scalar flag curvature. In Theorem 3.1 we show that the isotropy condition, together with three other conditions on the Jacobi endomorphism, characterize sprays that are metrizable by Finsler functions of scalar flag curvature. The proof of Theorem 3.1 provides an algorithm to construct the Finsler function of scalar flag curvature, in the case when a given spray is metrizable. One condition of Theorem 3.1, regarding the r…

Finsler 2-manifolds whose holonomy group is the diffeomorphism group of the circle

In this paper we show that the topological closure of the holonomy group of a certain class of projectively flat Finsler 2-manifolds of constant curvature is maximal, that is isomorphic to the connected component of the diffeomorphism group of the circle. This class of 2-manifolds contains the standard Funk plane of constant negative curvature and the Bryant-Shen-spheres of constant positive curvature. The result provides the first examples describing completely infinite dimensional Finslerian holonomy structures.

Finsler Metrics of Constant Positive Curvature on the Lie Group $s^3$

Guided by the Hopf fibration, we single out a family (indexed by a positive constant K) of right invariant Riemannian metrics on the Lie group $S^3$. Using the Yasuda-Shimada theorem as an inspiration, we determine for each K>1 a privileged right invariant Killing field of constant length. Each such Riemannian metric pairs with the corresponding Killing field to produce a y-global and {\it explicit} Randers metric on $S^3$. Using the machinery of spray curvature and Berwald's formula for it, we prove directly that the said Randers metric has constant positive flag curvature K, as predicted by …

特殊芬斯勒流形上的平行1-形式

Parallel one forms on special Finsler manifolds

本文研究特殊芬斯勒流形上平行1-形式的存在性，推导了Landsberg流形相关曲率秩的结论，证明了几类特殊芬斯勒度量均不容许平行1-形式。

关于几乎有理Finsler度量

On almost rational Finsler metrics

本文提出几乎有理Finsler度量概念，给出其成为Riemannian度量的充要条件，研究相关几何量的有理性，证明若干曲率条件下的刚性结果，并引入扩展m次根度量作为新例子。

关于广义$m$-克罗皮纳（$m$-Kropina）度量

On the generalized $m$-Kropina metrics

本文研究广义$m$-克罗皮纳度量的芬斯勒几何性质，证明其为几乎有理芬斯勒度量，推导其为里奇平坦度量的条件及刚性约束，得出其成为特定真空场方程精确解的条件，并给出四维实例。