2015 Fiscal Year Final Research Report
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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| Keywords | Riemannian manifolds / Finsler manifolds / geodesics / cut locus / convex functions / Busemann functions / Hausdorff dimension / fractals |
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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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| Free Research Field |
微分幾何学
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