The projective geometry of Zoll surfaces and the Cut locus on Finsler manifolds

在更一般的情况下，我们研究了Randers度量的割轨迹，而不是具有Killing向量场的Zermelo导航问题的解。实际上，Randers流形上的割轨迹的结构可以在没有任何曲率或Killing相关性质的情况下确定。这表明存在很大类Finsler度量，它们的割轨迹结构是可以确定的。第一步是将Finsler度量构造为Zermelo关于黎曼度量h和Killing域V的导航问题解的解，然后通过封闭的一形式来进行Beta变换。这种构造自然推广到Zermelo对(F，V)的导航的情况，其中F是先验给定的Randers型Finsler度量，V是F-Killing域。芬斯勒杀伤场的研究是现代芬斯勒几何学中一个复杂的课题。讨论了Finsler度量F的等距群与流形的上同调群的关系，并将所给出的构造推广到黎曼度量序列和Killing域序列的情形，得到了具有可计算测地线、曲率和割轨迹的新Finsler度量序列.这是现代芬斯勒几何学中一种全新的趋势，需要进一步关注。

We have studied the cut locus of Randers metrics in a more general case than the solutions of Zermelo's navigation problem with Killing vector fields. Indeed, the structure of cut locus on a Randers manifold can be determined without any curvature or Killing-related property. This shows that there are very large classes of Finsler metrics whose cut locus structure can be determined.The construction is done in 2 steps. First step is to construct Finsler metrics as solutions of Zermelo's navigation problem solution for the Riemannian metric h and a Killing field V, followed by a beta-change by means of a closed one-form. The construction naturally extends to the case of the Zermelo's navigation for (F,V), where F is an a-priori given Finsler metric of Randers type and V an F-Killing field. The study of Finsler Killing fields is a complex topic in modern Finsler gometry. The dimension of the isometry group of the Finsler metric F and the cohomology group of the manifold are related.Moreover, the construction given here was further generalized to the case of a sequence of Riemannian metrics and a sequence of Killing fields that leads to sequences of new Finsler metrics with computable geodesics, curvatures and cut loci. This is a completely new trend in modern Finsler geometry that needs further attention.