研究課題/領域番号 |
20K03595
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研究種目 |
基盤研究(C)
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配分区分 | 基金 |
応募区分 | 一般 |
審査区分 |
小区分11020:幾何学関連
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研究機関 | 東海大学 |
研究代表者 |
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研究期間 (年度) |
2020-04-01 – 2024-03-31
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研究課題ステータス |
交付 (2022年度)
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配分額 *注記 |
1,950千円 (直接経費: 1,500千円、間接経費: 450千円)
2022年度: 650千円 (直接経費: 500千円、間接経費: 150千円)
2021年度: 650千円 (直接経費: 500千円、間接経費: 150千円)
2020年度: 650千円 (直接経費: 500千円、間接経費: 150千円)
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キーワード | Finsler manifolds / geodesics / cut locus / isometry group / cohomology group / Zoll manifolds / Liouville manifolds / closed geodesics / cylinders of revolution / Randers metrics / conjugate locus / theory of geodesics / Riemannian manifolds / Cut Locus / Manifolds of geodesics |
研究開始時の研究の概要 |
I will study the projective geometry of the Zoll surfaces and the structure of the cut locus of Finsler manifolds, i.e. (a)the projective geometry of Zoll structures on spheres, and the geometry of the manifold of geodesics of a Zoll surface endowed with a Finsler structure of constant curvature. (b)the cut locus structure on special Finsler manifolds, and the relation of the cut locus with the geometrical and topological properties of these Finsler manifolds. In special, the structure of the cut locus of von-Mangoldt surfaces of Finsler type.
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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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現在までの達成度 (区分) |
現在までの達成度 (区分)
3: やや遅れている
理由
Due to the three years of Corona virus pandemic I was not able to do joint research as initially intended, nor to attend international conferences.
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今後の研究の推進方策 |
I intend to further develop geometrical methods to determine the structure of cut locus for Finsler manifolds more general than Randers metrics without any curvature restrictions. The sequence of Finsler metrics obtained will be further studied and the Gromov-Hausdorff type of convergence for Finsler metrics investigated. I also intend to further clarify the geometry of Finsler manifolds all of whose geodesics are closed and relation to Hamiltonian systems.
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