2020 Fiscal Year Research-status Report
The projective geometry of Zoll surfaces and the Cut locus on Finsler manifolds
| Project/Area Number |
20K03595
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| Research Institution | Tokai University |
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Principal Investigator |
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| Project Period (FY) |
2020-04-01 – 2023-03-31
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| Keywords | Finsler manifolds / theory of geodesics / Riemannian manifolds / conjugate locus / cut locus |
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| Outline of Annual Research Achievements |
In the academic year 2020 I have studied the cut locus structure of Finsler metrics of Randers type obtained as solutions of the Zermelo's navigation problem (h, W) on surfaces.
The navigation problem is to construct a Finsler metric of Randers type starting with a Riemannian metric h and a vector field W (called the wind). In the past we have studied the geodesics behavior and cut locus structure for such Randers metrics obtained in the case when the wind W is a Killing vector field with respect to the Riemannian metric h. In the present research I have studied the local and global behavior of geodesics in the case when the wind W is not a Killing vector field, but a more general one. By using a two-step construction we have succeeded to determine the properties of geodesics and the structure of the cut locus in the Finsler case generalizing in this way the results obtained in the past by us and other researchers.
The present results have been already submitted to an international journal. Some special cases including surfaces of revolution and other Finsler surfaces were also studied and the results included in other two papers under preparation.
I have also studied some basic properties of the geodesics of Finsler surfaces of Zoll type and the geometry of affine connections equivalence classes all of whose geodesics are closed. The geometry of Finsler manifolds endowed with metrics of Liouville type is a completely new concept for Finsler geometry that needs detailed research.
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| Current Status of Research Progress |
Current Status of Research Progress
3: Progress in research has been slightly delayed.
Reason
Due to the corona pandemic I wasn't able to attend any conferences abroad nor to visit other specialists in Japan or abroad for joint research.
The initial plan for 2020 contained two topics. First was to solve in 2020 the cut locus problem for more general Finsler manifolds than Randers one obtained from Zermelo's navigation problem with Killing vector field. I was able to research this topic and obtain results that worth publishing.
The second one was to study the projective geometry of Zoll metrics in two and three dimensions. Due to the reason above, that is the impossibility of traveling and doing joint research, this topic is slightly delayed.
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| Strategy for Future Research Activity |
I intend to stick to the initial Research Plan and do research in the following two directions:
1. Study the cut locus for Finsler manifolds more general than the Randers case obtained from Zermelo's navigation with Killing vector field. I intend to extend this type of research to von Mangoldt surfaces and Finsler-Liouville surfaces. I will extend these results to the study of Busemann functions on non-compact Finsler manifolds.
2. Study the projective geometry of Zoll metrics. This is the study of affine connections equivalence classes all of whose geodesics are closed. In this case also the relation with the Liouville manifolds will be considered.
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| Causes of Carryover |
Due to the corona pandemic I was not able to attend international conferences. In the case the pandemic will reach to an end I intend to attend several international conferences and to engage in several joint research projects.
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