Cut locus and variational problems with constaints on Finsler manifolds
| Project/Area Number |
17K05226
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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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| Research Field |
Geometry
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| Research Institution | Tokai University |
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Principal Investigator |
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| Project Period (FY) |
2017-04-01 – 2020-03-31
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| Project Status |
Completed (Fiscal Year 2019)
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| Budget Amount *help |
¥2,340,000 (Direct Cost: ¥1,800,000、Indirect Cost: ¥540,000)
Fiscal Year 2019: ¥780,000 (Direct Cost: ¥600,000、Indirect Cost: ¥180,000)
Fiscal Year 2018: ¥780,000 (Direct Cost: ¥600,000、Indirect Cost: ¥180,000)
Fiscal Year 2017: ¥780,000 (Direct Cost: ¥600,000、Indirect Cost: ¥180,000)
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| Keywords | Finsler manifolds / Riemannian manifods / geodesics / cut locus / variational problem / Quantum mechanics / Cut locus / Riemannian manifolds / Variational problem / constant flag curvature / Zoll metrics / distance function / Killing vectors / manifolds topology / surfaces of revolution / variational problems / Busemann function |
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| Outline of Final Research Achievements |
In this research I have studied the cut locus and variational problems on Finsler manifolds. Both of them are important problems in Differential Geometry, one of the most old and fundamental fields of modern mathematics.
Finsler manifolds are spaces where the geometry and Physics of the space depend on the direction. Therefore, distances between points, shortest paths (called geodesics) and many other geometrical properties depend on the direction. In the Euclidean space, distances between points are the same in one direction as well as in the oposite direction, however just imagine a strong wind blowing. If traveling with a constant speed engine, clearly we can regard the time needed to reach point B from a point A as the distance between the points A and B. The everyday life experience teaches that the time needed to travel from A to B is not the same as going back under a strong wind. Finsler geometry is a realistic model of the real World and this make it very important.
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| Academic Significance and Societal Importance of the Research Achievements |
本研究で、フィンスラー多様体の最小跡に関する新しい結果を得ることができました。最小跡というのは、距離関数は滑らかではなく、特異点を持っているところです。このような点を超えると、今まで最短だった道は最短性を失ってしまうので、非常に重要です。そのために、制御理論、様々な科学の分野、産業などにたくさん応用があります。
また、変分問題やフィンスラー多様体の幾何学との間の関係を明確にし、それを量子力学の分野に応用しました。これらの成果は数学や物理学の国際誌の投稿し、一部はすでに受理され、閲覧可能になっています。
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Report
(4 results)
Research Products
(12 results)
