The Cut Locus of Riemannian Manifolds: a Surface of Revolution
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Abstract
Abstract
This article reviews the structure theorems of the cut locus for very familiar surfaces of revolution. Some properties of the cut locus of a point of a Riemannian manifold are also discussed.
Keywords: Riemannian manifolds, surface of revolution, homeomorphic
*Corresponding author:
E-mail: tanaka@tokai-u.jp
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References
[2] Myers, S. B., 1936. Connections between differential geometry and topology. II. Closed surfaces, Duke Mathematics Journal, 2(1), 95-102.
[3] Gluck, H. and Singer, D., 1979. Scattering of geodesic fields. II, Ann.Math.110, 205-225.
[4] Hebda, J., 1994. Metric structure of cut loci in surfaces and Ambrose’s problem, Journal of Differential Geometry, 40, 621-642.
[5] Hartman, P., 1964. Geodesic parallel coordinates in the large, Amer. J. math. 86, 705-727.
[6] Shiohama, K., and Tanaka, M., 1993. The length function of geodesic parallelcircles, in Progress in Differential Geometry, The Mathematical Society of Japan, Tokyo, 299-308.
[7] Shiohama, K. and Tanaka, M., 1996. Cut loci and distance spheres on Alexandrov surfaces, Acte de la Table Ronde de Géometrie Différentielle (Luminy, (1992), 531-559, Sémin. Congr., 1, Société mathématique de France, Paris.
[8] 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.
[9] Itoh, J.I. and Tanaka, M., 1998. The dimension of a cut locus on a smooth Riemannian manifold, Tohoku Mathematics Journal, 50, 574-575.
[10] Elerath, D., 1980. An improved Toponogov comparison theorem for non-negatively curved manifolds, Journal of Differential Geometry, 15, 187-216.
[11] Itoh, J. and Kiyohara, K., 2004. The cut and the conjugate loci on ellipsoids, Manuscripta Matematics, 114, 247-264.
[12] Itoh, J. and Kiyohara, K., 2010. The cut loci on ellipsoids and certain Liouville manifolds, Asian Journal of Matematics, 14, 257-289.
[13] Itoh, J. and Kiyohara, K., 2011. Cut loci and conjugate loci on Liouville surfaces, Manuscripta Mathematics, 136, 115-141.
[14] Chitsakul, P., 2014. The structure theorem for the cut locus of a certain class of cylinders of revolution I. Tokyo Journal Mathematics, 37, 473-484
[15] Chitsakul, P., 2015. The structure theorem for the cut locus of a certain class of cylinders of revolution II. Tokyo Journal Mathematics, 38, 239-248
[16] Gravesen, J., Markvorsen, S., Robert, S. and Tanaka, M., 2005. The cut locus of a torus of revolution, Asian Journal of Mathematics, 9, 103-120.