The projective geometry of Zoll surfaces and the Cut locus on Finsler manifolds
| Project/Area Number |
20K03595
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| Research Category |
Grant-in-Aid for Scientific Research (C)
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| Allocation Type | Multi-year Fund |
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| Section | 一般 |
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| Review Section |
Basic Section 11020:Geometry-related
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| Research Institution | Tokai University |
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Principal Investigator |
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| Project Period (FY) |
2020-04-01 – 2024-03-31
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| Project Status |
Completed (Fiscal Year 2023)
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| Budget Amount *help |
¥1,950,000 (Direct Cost: ¥1,500,000、Indirect Cost: ¥450,000)
Fiscal Year 2022: ¥650,000 (Direct Cost: ¥500,000、Indirect Cost: ¥150,000)
Fiscal Year 2021: ¥650,000 (Direct Cost: ¥500,000、Indirect Cost: ¥150,000)
Fiscal Year 2020: ¥650,000 (Direct Cost: ¥500,000、Indirect Cost: ¥150,000)
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| Keywords | Finsler manifolds / Theory of geodesics / cut locus / Zoll metrics / the navigation problem / closed geodesics / manifold of geodesics / flag curvature / geodesics / isometry group / cohomology group / Zoll manifolds / Liouville manifolds / cylinders of revolution / Randers metrics / conjugate locus / theory of geodesics / Riemannian manifolds / Cut Locus / Manifolds of geodesics |
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| Outline of Research at the Start |
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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| Outline of Annual Research Achievements |
In the present research proposal I have included twotopics: 1. the study the geometrical and topological properties of Finsler metrics of constant positive flag curvature induced by Zoll metrics, and 2. the study the geometry and topology of Finsler manifolds by using the properties of distance function and the cut locus. We have obtained interesting results concerning both topics, some of them published already, others still in print.
1.We have determined the local and global behaviour of geodesics, the difference with the Riemannian case and the structure of the cut locus of a point.We have determined the deviation of the Riemannian geodesics and the deformation of the Riemannian cut locus. We have performed numerical simulation on computer using the programming language SAGE in order to draw explicitly the trajectories of some geodesics and of the cut locus in the Finslerian case.
2. We have studied the cut locus of Randers type metrics on different surfaces of revolution we have determined the local and global behaviour of geodesics as well as the structure of the cut locus. In this research we have developed the Hamiltonian formalism for Randers metrics, all computation being index free on the cotangent bundle. During this research we have discovered that the geodesics behaviour and the structure of the cut locus can be explicitly determined in a much more general case than the Zermelo's navigation case with Killing wind, as known previously. The relation with the Finsler metrics of scalar curvature was also investigated in the case of 3-manifolds.
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Report
(4 results)
Research Products
(7 results)
