| Project/Area Number |
15K04864
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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 | Fukuoka Institute of Technology |
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Principal Investigator |
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| Co-Investigator(Kenkyū-buntansha) |
永野 哲也 長崎県立大学, 情報システム学部, 教授 (00259699)
糸川 銚 福岡工業大学, 情報工学部, 教授 (90223205)
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| Project Period (FY) |
2015-04-01 – 2019-03-31
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| Project Status |
Completed (Fiscal Year 2018)
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| Budget Amount *help |
¥4,160,000 (Direct Cost: ¥3,200,000、Indirect Cost: ¥960,000)
Fiscal Year 2018: ¥910,000 (Direct Cost: ¥700,000、Indirect Cost: ¥210,000)
Fiscal Year 2017: ¥1,040,000 (Direct Cost: ¥800,000、Indirect Cost: ¥240,000)
Fiscal Year 2016: ¥1,040,000 (Direct Cost: ¥800,000、Indirect Cost: ¥240,000)
Fiscal Year 2015: ¥1,170,000 (Direct Cost: ¥900,000、Indirect Cost: ¥270,000)
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| Keywords | フィンスラー多様体 / 測地線 / 切断跡 / 共役跡 / 凸性 / 非対称距離構造 / 指数写像 / 非対称距離関数 / 凸性と凸関数 / ランダース計量 / フィンスラー幾何学 / 凸関数と凸集合 |
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| Outline of Final Research Achievements |
The notion of convexity is very important in the study of natural science including mathematics and engineering. Finsler manifolds are considered as a natural extension of Riemannian manifolds. In particular, Finsler manifolds admit non-symmetric metric structure. This point is important for our investigation. The behavior of geodesics on Finsler manifolds is completely different from those of Riemannian manifolds. This is because of the property of the fundamental function.
We have investigated the strucure of cut loci and conjugate loci on Finsler manifolds. In particular, the convexity of Finsler manifolds is one of the main them of our investigation. Through our investigation, we have discovered new phenomena on the behavior of geodesics on Finsler manifolds, which have never seen on Riemannian world. Our main results have been in public from Pacific J. Math., Trans. Amer. Math. Soc., Math. Debrecen, Manuscripta Math.
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| Academic Significance and Societal Importance of the Research Achievements |
従来のリーマン幾何学では捉える事が出来なかった非対称距離構造の研究はフィンスラー幾何学の特徴を示している．フィンスラー多様体上の２点間の距離が往路と復路では異なると言う点において，フィンスラー幾何学は現実の社会に即していると考えられる．特に，測地線の大域的挙動を調べる事が重要な研究課題となる．測地線の大域的研究に重要な役割を果たす切断跡，共役跡に関する基本的な性質をフィンスラー多様体上で研究した．研究成果は米国，ドイツ，印度，ハンガリー等の学術雑誌から発表されている．従って，今後の研究の指針がこれらの成果から示され各国でフィンスラー幾何学の研究が進むであろう．
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