| 研究課題/領域番号 |
17K05226
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| 研究機関 | 東海大学 |
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研究代表者 |
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| 研究期間 (年度) |
2017-04-01 – 2020-03-31
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| キーワード | Riemannian manifolds / Finsler manifolds / geodesics / distance function / Killing vectors / cut locus / manifolds topology / surfaces of revolution |
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| 研究実績の概要 |
As described in the initial research plan, in the 1-st year, I have concentrate my research on the behaviour of geodesics and the cut locus for Finsler manifolds.
In particular I have determined the properties of geodesics and cut locus for some Finsler metrics obtained by deformation of Riemannian metrics by means of a linear one-form. I have determined the precise geodesics behaviour for a Randers metric globally defined on the two sphere of revolution, of a global Kropina metric defined on those manifolds that admit a unit Killing vector field (some topological restrictions imposed) and of the slope metric on Finsler surfaces of revolution. These results were summarized in 3 different papers already submitted for referee to some international journals. Two of them were already accepted with minor revisions, I am waiting for the referee report for the third one.
The second topic is about variational problem on Finsler manifolds. I have clarified the derivaton of Euler-Lagrange equations in the case of Finsler functionals and of lifts to the tangent bundle. These equations are essential for the research following. I intend to summarize these results in a paper and submit for referee 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
I think the present research is progressing rather smoothly because I was able to accomplish the initial research plan for the first year. In particular, determining the cut locus of special Finsler manifolds like Randers, Kropina or slope metrics and to clarify the derivation of Euler-Lagrange equations from the variational problem for Finsler manifolds.
Using these results I can move on to the study of Busemann function and specific variational problems for Riemannian and Finsler manifolds. In special I am interested in the study of the variational problem on Finsler manifolds where the functional is determined by the geodesic curvature of a Riemannian or Finsler metric. This type of variational problem can be studied with constraints and without constraints.
Another important case is the case when integrating with respect to the volume form. This type of variational problems is related to the notion of Einstein manifolds and Einstein equations in Finsler spaces.
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| 今後の研究の推進方策 |
In the future, I will continue the research of cut locus and variational problems like in the original research plan. The main topics to research in this year are the variational problems for the functional given by the integral of the geodesic curvature and the functionals given by scalar curvatures with respect to the volume forms.
I also expect to finish soon a paper about the existence of convex functions on Finsler manifolds and show how to generalize the Riemannian conditions for the existence of this function.
Finaly, I intend to extend my research to the Busemann functions and in special to the study of the copoints set on Riemannian and Finsler manifolds.
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| 次年度使用額が生じた理由 |
スケジュールの調整がつかず予定していた出張をキャンセルした
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