| 研究実績の概要 |
1. The existence of Riemannian and Finsler structures whose cut locus is a fractal. 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. The geometry and topology of Finsler manifolds admitting convex functions
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 started the study of Busemann functions on Finsler manifolds, and have obtained some preliminary results on the convexity of Busemann functions on Finsler manifolds. Unfortunately, the condition for the convexity of Busemann functions on Finsler manifolds is more complicated than in the Riemannian geometry. Indeed, the flag curvature alone is not able to control the conbvexity of Busemann functions on Finsler manifolds.
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