| 研究実績の概要 |
In 2019 I focused my research in clarifying the geometry of Finsler metrics of constant flag curvature K=1 on the sphere, the geometry of geodesics and relation to the cut locus. In a joint research with Prof. K. Kiyohara and K. Shibuya we were able to construct explicitely Finsler metrics of constant flag curvature on the manifold of geodesics of a Zoll metric. These Finsler metrics are more general those constructed by R. Bryant in the past. The geodesics and cut locus were also explicitely determined. These results appeared in a research paper published in an international journal.
Another topic that I worked is related to the variational problem in Finsler spaces. In a joint work with Prof. S. Liang and T. Harko, we have succeeded in extending the classical approach for the calculus of variations for time depending Lagrangians to the Finslerian case. Indeed, any variational problem for a time depending Lagrangian can be regarded as a variational problem for Finsler metrics of Kropina type. Even though these are not classical Finsler metrics, but conic ones, we were able to construct the precise relation between the geodesics and other geometrical features of the variational problem in terms of the geometry of Finsler metrics, including some interesting results about geodesics and cut locus. Finally, we have applied these results to the famous problem of geometrization of Quantum mechanics. These results also appeared in an international journal.
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