The Theory of Geodesics on Some Surface of Revolution

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Published: Jan 5, 2018
Keywords:
Randers rotationalsphere, surface of revolution, Zermelo navigation

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Rattanasak Hama*
Department of Mathematics, Faculty of Science, King Mongkut’s Institute of Technology Ladkrabang, Bangkok, Thailand
Jaipong Kasemsuwan
Department of Mathematics, Faculty of Science, King Mongkut’s Institute of Technology Ladkrabang, Bangkok, Thailand

Abstract

We study the properties of the geodesics on a Randers rotational surface of revolution by using Zermelo navigation data (h,W), where h is the induced Riemannian metric on the surface of revolution and W is the rotational wind. We are in special interested in the half-period function that can be computed by similar methods to the Riemannian case. Our result can be applied to find the structure of the cut locus of a Randers rotational 2-sphere of revolution.


Keywords:Randers rotationalsphere, surface of revolution, Zermelo navigation


*Corresponding author:      


E-mail: [email protected]

Article Details

Issue
Vol. 17 No. 1 (2017)
Section
Original Research Articles

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References

[1] Bao, D., Chern, S.S. and Shen, Z., 2000. An Introduction to Riemann Finsler Geometry, Springer, GTM 200.
[2] Bao, D., Robles, C., 2004. Ricci and flag curvatures in Finsler geometry Riemann-Finsler Geometry. MSRI Publications50, 198-256.
[3] Shiohama, K., Shioya, T. and Tanaka, M., 2003. The Geometry of Total Curvature on Complete Open Surfaces. Cambridge tracts in mathematics 159, Cambridge University Press, Cambridge.
[4] Bao, D., Robles, C., Shen, Z., 2004. Zermelo Navigation on Riemannian manifolds. Journal of Differential Geometry, 66, 377-435.
[5] Hama, R., Chitsakul, P. and Sabau, S.V., 2015. The Geometry of a Randers Rotational Surface. Publicationes Mathematicae Debrecen, 87(3-4), 473-502.
[6] Robles, C., 2007. Geodesics in Randers Spaces of Constant Curvature. Transactions of the American mathematical society, 359(4), 1633-1651.
[7] Sinclair, R. and Tanaka, M., 2007. The cut locus of a two sphere of revolution and toponogov's comparison theorem. Tohoku Math. J., 59, 379-399.