| 研究課題/領域番号 |
25400075
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| 研究種目 |
基盤研究(C)
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| 研究機関 | 東海大学 |
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研究代表者 |
SABAU Vasile Sor 東海大学, 札幌教養教育センター, 准教授 (80364280)
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| 研究期間 (年度) |
2013-04-01 – 2016-03-31
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| キーワード | Riemannian manifolds / Finsler manifolds / cut locus / convex functions / Busemann functions / topological cylinder / level sets' structure / differentiability |
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| 研究概要 |
In this year, we have revised and finalized the construction of a Riemannian metric on the Sphere and a Finsler metric of Randers type whose cut locus is a fractal. This is a joint research with Professor J. Itoh from Kumamoto University.These metrics are not smooth metrics, but only k-class metrics. We have summarized these findings in a paper and have submitted already to an international journal.
In a joint research with Professor K. Shiohama from Fukuoka University we have studied the geometrical properties of convex functions defined on Finsler manifolds. Moreover, the structure of the level sets of convex functions as well as the topological restrictions of the existence of convex functions on Finsler manifolds were thoroughly investigated. These results were summarized in a research paper and submitted to an international journal.
Moreover, I have started the study of Busemann functions on a Finsler manifold and obtain first results about the differentiability of this kind of functions as well as about the structure of the co points set. I intend to summarize these findings in a paper and submit it to an international journal in the near future.
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| 現在までの達成度 (区分) |
現在までの達成度 (区分)
2: おおむね順調に進展している
理由
We consider that the research is progressing smoothly according to the research plan. Actually, I have interchanged one topic, I proposed to study the Busemann functions in 2013 and the properties of convex functions on Finsler manifolds in 2014, but I studied the properties of convex functions on Finsler manifolds in 2013 and leave the study of the Busemann functions for 2014.
The reason of doing this is because the definition of convex functions on Finsler manifolds was more complex as initially thought and complete classification of the topological restrictions on the topology of the base manifold took a lot of time. However, this interchange of topics is beneficial for the present research because it provides general results about the behavior of convex functions on Finsler manifolds, results that can be applied to the case of Busemann functions on some Finsler manifolds of non-negative flag curvature.
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| 今後の研究の推進方策 |
I intend to stick with the research plan described in the application form.
More specifically, in this year I intend to study the properties of cut locus of a Finsler manifold and compare these with the similar results for Riemannian manifolds.
Moreover, I will focus in the study of geometrical properties of the Busemann functions on Finsler manifolds and the structure of the co-points set. We expect that this research will give similar results as in the Riemannian case. However, the fact that the Finslerian geodesics are not reversible and that the Finslerian distance function is not symmetric needs a special treatment. Even the final results are similar with the Riemannian counterpart, the arguments are quite different.
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| 次年度の研究費の使用計画 |
外国書籍を購入する予定だったのですが、注文が間に合わなかったため、次年度使用を予定しております
外国書籍を購入
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