2018 Fiscal Year Research-status Report
Cut locus and variational problems with constaints on Finsler manifolds
| Project/Area Number |
17K05226
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| Research Institution | Tokai University |
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Principal Investigator |
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| Project Period (FY) |
2017-04-01 – 2020-03-31
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| Keywords | Finsler manifolds / Riemannian manifolds / constant flag curvature / Zoll metrics |
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| Outline of Annual Research Achievements |
In 2018 I have studied in principal the properties of Finsler manifolds of constant flag curvature induced by a Zoll metric. This is strictly related to the behavior of geodesics and the variational problem in Riemannian and Finsler manifolds. I was prompted to this topic when working on the variational problem of the squared geodesic curvature, topic included in the original research plan.
Another topic that I worked in 2018 was on the variational problem of first order and higher order with and without constrains. Our basic result here is that the variational problem for any Lagrangian can be studied as variational problem for some special classes of Finsler metrics. In this way, we show the universality of Finsler metrics and of the Finslerian theory of geodesics.
We have applied these findings to several problems in classical mechanics and in quantum mechanics.
In the study of variational problem it becomes clear that the classical Lagrangian approach is too complicated to give geometrically meaningful results, however the Hamiltonian approach is much more suitable for this kind of research, so we have started to study Finsler manifolds using Hamiltonian formalism. This approach pointed us to the study of Integrable systems on Finsler manifolds, a completely new topic. Integrable systems are very well studied on Riemannian manifolds bu the study in the Finsler case is completely new.
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| Current Status of Research Progress |
Current Status of Research Progress
3: Progress in research has been slightly delayed.
Reason
The progress of the present research is slightly delayed because of the unexpected complexity of the Finsler metrics of constant flag curvature. Likewise, the finding that we can express the variational problem for any Lagrangian in terms of the geodesics theory for some special Finsler manifolds lead us to new and unexpected results that deserve more attention. We intend to publish several papers (some of them already submitted to international journals in 2018) about these new findings that we could not infer at the beginning of this research.
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| Strategy for Future Research Activity |
In the future, I intend to stick to the original research plan and to clarify all the details I can about the behavior of geodesics and cut locus of Finsler metrics of constant flag curvature, relation with Busemann functions and so on.
On the other hand, I will concentrate my effort to the study of variational problems by using the new approach that we have discovered in 2018.
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Research Products
(3 results)
