Characterization of Curves that Lie on a Geodesic Sphere or on a Totally Geodesic Hypersurface in a Hyperbolic Space or in a Sphere
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The consideration of the so-called rotation minimizing frames allows for a simple and elegant characterization of plane and spherical curves in Euclidean space via a linear equation relating the coefficients that dictate the frame motion. In this work, we extend these investigations to characterize curves that lie on a geodesic sphere or totally geodesic hypersurface in a Riemannian manifold of constant curvature. Using that geodesic spherical curves are normal curves, i.e., they are the image of an Euclidean spherical curve under the exponential map, we are able to characterize geodesic spherical curves in hyperbolic spaces and spheres through a non-homogeneous linear equation. Finally, we also show that curves on totally geodesic hypersurfaces, which play the role of hyperplanes in Riemannian geometry, should be characterized by a homogeneous linear equation. In short, our results give interesting and significant similarities between hyperbolic, spherical, and Euclidean geometries.
KeywordsRotation minimizing frame geodesic sphere spherical curve hyperbolic space sphere totally geodesic submanifold
Mathematics Subject Classification53A04 53A05 53B20 53C21
The authors would like to thank Gilson S. Ferreira-Júnior and Gabriel G. Carvalho for useful discussions, the anonymous Referees for their suggestions which have improved the quality of the text, and also the financial support provided by Conselho Nacional de Desenvolvimento Científico e Tecnológico-CNPq (Brazilian agency).
- 3.Bloomenthal, J., Riesenfeld, R.F.: Approximation of sweep surfaces by tensor product NURBS. In: Silbermann, M.J., Tagare, H.D., (eds.) SPIE Proceedings, Curves and Surfaces in Computer Vision and Graphics II, vol. 1610, pp. 132–154. International Society for Optics and Photonics (1991)Google Scholar
- 11.da Silva, L.C.B.: Characterization of spherical and plane curves using rotation minimizing frames (2017). arXiv:1706.01577v3
- 14.Etayo, F.: Rotation minimizing vector fields and frames in Riemannian manifolds. In: Castrillón López, M., Hernández Encinas, L., Martínez Gadea, P., Rosado María, M.E. (eds.) Geometry, Algebra and Applications: From Mechanics to Cryptography, Springer Proceedings in Mathematics and Statistics, vol. 161, pp. 91–100. Springer, Berlin (2016)Google Scholar
- 22.Kühnel, W.: Differentialgeometrie: Kurven–Flächen–Mannigfaltigkeiten 5. Auflage. Vieweg+Teubner (2010)Google Scholar
- 25.Murphy, T., Wilhelm, F.: Random manifolds have no totally geodesic submanifolds. https://arxiv.org/abs/1703.09240 (To appear in Michigan Math. J.)
- 33.Wang, W., Jüttler, B., Zheng, D., Liu, Y.: Computation of rotation minimizing frames. ACM Trans. Graph. 27, Article 2 (2008)Google Scholar