The Theory of Geodesics on Some Surface of Revolution

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Rattanasak Hama*
Jaipong Kasemsuwan

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: rattanasakhama@gmail.com

Article Details

Section
Original Research Articles

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.