Geometry of geodesics of Riemannian and Finsler manifolds
| Project/Area Number |
25400075
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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) |
2013-04-01 – 2016-03-31
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| Project Status |
Completed (Fiscal Year 2015)
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| Budget Amount *help |
¥2,340,000 (Direct Cost: ¥1,800,000、Indirect Cost: ¥540,000)
Fiscal Year 2015: ¥780,000 (Direct Cost: ¥600,000、Indirect Cost: ¥180,000)
Fiscal Year 2014: ¥780,000 (Direct Cost: ¥600,000、Indirect Cost: ¥180,000)
Fiscal Year 2013: ¥780,000 (Direct Cost: ¥600,000、Indirect Cost: ¥180,000)
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| Keywords | Riemannian manifolds / Finsler manifolds / geodesics / cut locus / convex functions / Busemann functions / Hausdorff dimension / fractals / flag curvature / level sets / distance function / topological cylinder / level sets' structure / differentiability |
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| Outline of Final Research Achievements |
1.I have constructed (joint research with J. Itoh) Riemannian and Finslerian structures on spheres whose cut locus of a point is a fractal (i.e. the Hausdorff dimension of the cut locus is not integer). This result is interesting not only for Finsler geometry, but also for Riemannian geometry and it is in the same time consistent with the result of Itoh-Tanaka about the Hausdorff dimension of the cut locus of a smooth Riemannian manifold. Indeed, our Riemannian structure is not a smooth one.
2.I have introduced and studied the notion of convex functions on Finsler manifolds (joint research with K. Shiohama). Similarly with the Riemannian case, we have shown that there are topological restrictions for Finsler manifolds that admit convex functions. The difference with the Riemannian case was also clarified, as well as the influence of non-reversibility of geodesics in the Finslerian setting.As application, I have studied the convexity of Busemann functions on Finsler manifolds.
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Report
(4 results)
Research Products
(18 results)
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[Presentation] Convexity on Finsler manifolds2015
Author(s)
S. V. Sabau
Organizer
The 7-th International Congress of Romanian mathematicians
Place of Presentation
Al. I. Cuza Univ （Iasi, Romania）
Year and Date
2015-06-26
Related Report
Int'l Joint Research / Invited
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