| 研究実績の概要 |
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.
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