2014 Fiscal Year Research-status Report
Geometry of geodesics of Riemannian and Finsler manifolds
| Project/Area Number |
25400075
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| Research Institution | Tokai University |
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Principal Investigator |
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| Project Period (FY) |
2013-04-01 – 2016-03-31
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| Keywords | Finsler manifolds / Riemannian manifolds / Busemann functions / cut locus / convex functions / flag curvature / level sets / distance function |
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| Outline of Annual Research Achievements |
In this year I have extensively studied the Busemann functions on forward completely, non-compact Finsler manifold. In special I have clarified the relation between the set of co-points and the cut locus of the level set of a Busemann function. This research provides new results not only for Finsler manifolds, but also for Riemannian manifolds as well.
In the quest of concrete examples of Busemann functions I have considered Finsler surfaces of revolution. I have studied the differential geometry of these surfaces in order to clarify the geodesics behavior, cut locus, poles, rays and behavior of Busemann functions.
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| Current Status of Research Progress |
Current Status of Research Progress
3: Progress in research has been slightly delayed.
Reason
The research is progressing slightly delayed than initially planned because the research of Busemann functions on complete, non-compact Finsler manifolds took much more time than I thought. Indeed, the facts that the geodesics are not reversible and that the flag curvature, the Finslerian equivalent of the sectional curvature, lives on TM and not on the base manifold M bring challenging difficulties. Finally, almost all difficulties were surmounted, but this took little more time that expected.
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| Strategy for Future Research Activity |
I intend to catch up the delay occurred from the reasons explained above and to realize the initial proposed research plan. The topics remained are the conditions for a Finsler manifold to have cut locus with integer Hausdorff dimension, convexity of the Busemann function and implications of the existence of convex functions with compact levels on the isometry groups properties of a Finsler manifold. I intend to study thoroughly these topics in 2015.
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| Causes of Carryover |
購入予定の専門書は受注できなくて科研費使用可能な期間に間に合わなかったため
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| Expenditure Plan for Carryover Budget |
書籍を購入すること
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Research Products
(6 results)
