2021 Fiscal Year Research-status Report
The projective geometry of Zoll surfaces and the Cut locus on Finsler manifolds
| Project/Area Number |
20K03595
|
|---|
| Research Institution | Tokai University |
|---|
Principal Investigator |
|
|---|
| Project Period (FY) |
2020-04-01 – 2023-03-31
|
| Keywords | Finsler manifolds / Zoll manifolds / Liouville manifolds / closed geodesics / cut locus / cylinders of revolution / Randers metrics / conjugate locus |
|---|
| Outline of Annual Research Achievements |
In 2021 I have studied Zoll surfaces from Finslerian point of view. We succeeded in constructing Finsler metrics of Liouville type on Finsler surfaces all of whose geodesics are closed. This type of Finsler structures are completely new and they do not belong to any classical type of Finsler manifolds like Randers, Kropina or other known types. These Finsler metrics of Liouville type are natural generalizations of the Riemannian Liouville structures studied by Prof. K. Kiyohara in the past. The main problems appearing here are the positive homogeneity of the metrics and the positive definiteness of the Hessian matrix. Over all, we are developing the basics of Finsler geometry based on the study of the Hamiltonian rather than the Lagrangian.
I have also studied the second topic in the initial research plan, namely the cut locus of Finsler manifolds. We were not able to obtain new results about the cut locus of Finsler manifolds of other type than Randers type, but we have obtained remarkable progress concerning the study of the cut locus on topological cylinders endowed with a special type of Randers metric. We were lead in this way to some new classes of cylinders of revolution with remarkable geometrical properties in both perspectives: Riemannian and Finslerian ones. A particular type of Riccati equation is involved and we are considering a mixed approach than combines differential geometry with ODEs.
|
| Current Status of Research Progress |
Current Status of Research Progress
3: Progress in research has been slightly delayed.
Reason
The present research is slightly delayed due to the corona pandemic because I was not able to visit and do joint research as intended with my Research collaborators Prof. K. Kiyohara and K. Shibuya. We have collaborated through e-mail and Zoom, but not enough.
However, I think that the research in the present proposal is advancing constantly and that we will be able to achieve the intended purpose in the end.
|
| Strategy for Future Research Activity |
For the future, I intend to stick to the original research plan and to study more profoundly the Liouville type Finsler metrics we just have discovered in the last year. They are of Zoll type having many geometrical important properties.
I also intend to continue the study of the cut locus for Finsler surfaces using more general metrics than Randers metric, or, at least more general methods than Zermelo's navigation problem. In intend to obtain some new results on the Busemann functions on non-compact Finslerian manifolds.
|
| Causes of Carryover |
新型コロナウイルスの影響を受け、予定していた出張、学会等に参加できず予算不使用となったため。
|
