2022 Fiscal Year Research-status Report
The projective geometry of Zoll surfaces and the Cut locus on Finsler manifolds
Project/Area Number |
20K03595
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Research Institution | Tokai University |
Principal Investigator |
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Project Period (FY) |
2020-04-01 – 2024-03-31
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Keywords | Finsler manifolds / geodesics / cut locus / isometry group / cohomology group |
Outline of Annual Research Achievements |
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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Current Status of Research Progress |
Current Status of Research Progress
3: Progress in research has been slightly delayed.
Reason
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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Strategy for Future Research Activity |
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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Causes of Carryover |
Due to the corona virus pandemic I was not able to attend international conferences
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