Some remarks on Finsler manifolds with constant flag curvature
Some remarks on Finsler manifolds with constant flag curvature
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发表时间：
2001-07
影响因子：
0.3
通讯作者：
R. Bryant
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作者：
R. Bryant
本文是关于具有常正旗曲率的Finsler流形几何的四个松散相关的注记。第一点是，在这种流形的测地线空间上存在典型的Kahler结构。第二个注记是，在CPN中的适当一般位置的超曲面上，有一种自然的方法可以构造一个(不一定是完全的)常正旗曲率的Finsler n-流形。第三，在测地线空间上，用黎曼度量和1-形式刻画了S2上的常曲率Finsler度量。特别地，这使得人们可以使用S2上任意正高斯曲率的(黎曼)Zoll度量来构造S2上常正曲率的全局Finsler度量。第四个注记是关于n+1>；2维常正旗曲率的(局部)Finsler度量空间的一般性。证明了这种度量依赖于n+1个变量的n(n+1)个任意函数，并且这种度量自然地对应于2n-流形上的某些无挠的S1·GL(n，R)结构。作为副产品，我们发现这些群确实是作为2n维无扭仿射联络的完整系统出现的，这是迄今为止尚未发现的现象。1991年数学科目分类。53B40、53C60、58A15。
This article is an exposition of four loosely related remarks on the geometry of Finsler manifolds with constant positive flag curvature. The first remark is that there is a canonical Kahler structure on the space of geodesics of such a manifold. The second remark is that there is a natural way to construct a (not necessarily complete) Finsler n-manifold of constant positive flag curvature out of a hypersurface in suitably general position in CPn. The third remark is that there is a description of the Finsler metrics of constant curvature on S2 in terms of a Riemannian metric and 1-form on the space of its geodesics. In particular, this allows one to use any (Riemannian) Zoll metric of positive Gauss curvature on S2 to construct a global Finsler metric of constant positive curvature on S2. The fourth remark concerns the generality of the space of (local) Finsler metrics of constant positive flag curvature in dimension n+1 > 2. It is shown that such metrics depend on n(n+1) arbitrary functions of n+1 variables and that such metrics naturally correspond to certain torsion-free S1·GL(n,R)structures on 2n-manifolds. As a by-product, it is found that these groups do occur as the holonomy of torsion-free affine connections in dimension 2n, a hitherto unsuspected phenomenon. 1991 Mathematics Subject Classification. 53B40, 53C60, 58A15.