Randers metrics on two-spheres of revolution with simple cut locus

Rattanasak Hama; Sorin V. Sabau; Rattanasak Hama; Sorin V. Sabau

doi:10.3934/math.20231337

AIMS Mathematics

2023, Volume 8, Issue 11: 26213-26236. doi: 10.3934/math.20231337

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Research article

Randers metrics on two-spheres of revolution with simple cut locus

Rattanasak Hama ^{1
,
,},
Sorin V. Sabau ²

1.
Faculty of Science and Industrial Technology, Prince of Songkla University, Surat Thani Campus, Surat Thani 84000, Thailand
2.
Graduate School of Science and Technology, Physical and Mathematical Sciences, Tokai University, Sapporo 005-8600, Japan

Received: 06 July 2023 Revised: 18 August 2023 Accepted: 18 August 2023 Published: 13 September 2023
MSC : 53C22, 53C60

In the present paper, we study the Randers metric on two-spheres of revolution in order to obtain new families of Finsler of Randers type metrics with simple cut locus. We determine the geodesics behavior, conjugate and cut loci of some families of Finsler metrics of Randers type whose navigation data is not a Killing field and without sectional or flag curvature restrictions. Several examples of Randers metrics whose cut locus is simple are shown.
- Riemannian manifolds,
- geodesics,
- conjugate locus,
- two-sphere of revolution,
- Randers metric,
- Finslers metric,
- cut locus
Citation: Rattanasak Hama, Sorin V. Sabau. Randers metrics on two-spheres of revolution with simple cut locus[J]. AIMS Mathematics, 2023, 8(11): 26213-26236. doi: 10.3934/math.20231337

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Abstract

In the present paper, we study the Randers metric on two-spheres of revolution in order to obtain new families of Finsler of Randers type metrics with simple cut locus. We determine the geodesics behavior, conjugate and cut loci of some families of Finsler metrics of Randers type whose navigation data is not a Killing field and without sectional or flag curvature restrictions. Several examples of Randers metrics whose cut locus is simple are shown.

References

[1]	D. Bao, C. Robles, Ricci and flag curvatures in Finsler geometry, In: A sampler of finsler geometry, Cambridge: Cambridge University Press, 50 (2004), 197–259.
[2]	D. Bao, C. Robles, Z. Shen, Zermelo navigation on Riemannian manifolds, J. Differential Geom., 66 (2004), 377–435.
[3]	D. Bao, S. S. Chern, Z. Shen, An introduction to Riemann Finsler geometry, Springer Science & Business Media, 2000.
[4]	A. L. Bess, Manifolds all of whose Geodesics are Closed, In: Ergebnisse des Mathematik und ihrer Grenzgebiete, Berlin: Springer-Verlag, 1978.
[5]	B. Bonnard, J. B. Caillau, R. Sinclair, M. Tanaka, Conjugate and cut loci of a two-sphere of revolution with application to optimal control, Ann. Inst. H. Poincaré Anal. Non Linéaire, 26 (2009), 1081–1098. https://doi.org/10.1016/j.anihpc.2008.03.010 doi: 10.1016/j.anihpc.2008.03.010
[6]	P. Foulon, V. S. Matveev, Zermelo deformation of Finsler metrics by Killing vector fields, Electron. Res. Announc. Math. Sci., 25 (2018), 1–7. https://doi.org/10.3934/era.2018.25.001 doi: 10.3934/era.2018.25.001
[7]	R. Hama, J. Kasemsuwan, S. V. Sabau, The cut locus of a Randers rotational 2-sphere of revolution, Publ. Math. Debrecen, 93 (2018), 387–412.
[8]	R. Hama, S. V. Sabau, The geometry of a Randers rotational surface with an arbitrary direction wind, Mathematics, 8 (2020), 2047. https://doi.org/10.3390/math8112047 doi: 10.3390/math8112047
[9]	R. Miron, D. Hrimiuc, H. Shimada, S. V. Sabau, The geometry of Hamilton and Lagrange spaces, Springer Science & Business Media, 2001.
[10]	G. Randers, On an asymmetric metric in the four-space of general relativity, Phys. Rev., 59 (1941), 195–199. https://doi.org/10.1103/PhysRev.59.195 doi: 10.1103/PhysRev.59.195
[11]	C. Robles, Geodesics in Randers spaces of constant curvature, Trans. Amer. Math. Soc., 359 (2007), 1633–1651.
[12]	S. V. Sabau, M. Tanaka, The cut locus and distance function from a closed subset of a Finsler manifold, Houston J. Math., 42 (2016), 1157–1197.
[13]	Z. Shen, Finsler metrics with K = 0 and S = 0, Canad. J. Math., 55 (2003), 112–132.
[14]	K. Shiohama, T. Shioya, M. Tanaka, The geometry of total curvature on complete open surfaces, Cambridge: Cambridge University Press, 2003.
[15]	R. Sinclair, M. Tanaka, The cut locus of a two-sphere of revolution and Toponogov's comparison theorem, Tohoku Math. J., 59 (2007), 379–399. https://doi.org/10.2748/tmj/1192117984 doi: 10.2748/tmj/1192117984
[16]	M. Tanaka, T. Akamatsu, R. Sinclair, M. Yamaguchi, A new family of latitudinally corrugated two-spheres of revolution with simple cut locus, Tokyo J. Math., 46 (2023), 33–45. https://doi.org/10.3836/tjm/1502179366 doi: 10.3836/tjm/1502179366

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