Characterization of Curves that Lie on a Geodesic Sphere or on a Totally Geodesic Hypersurface in a Hyperbolic Space or in a Sphere

  • Luiz C. B. da Silva
  • José Deibsom da Silva
Article

Abstract

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.

Keywords

Rotation minimizing frame geodesic sphere spherical curve hyperbolic space sphere totally geodesic submanifold 

Mathematics Subject Classification

53A04 53A05 53B20 53C21 

Notes

Acknowledgements

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).

References

  1. 1.
    Benedetti, R., Petronio, C.: Lectures on Hyperbolic Geometry. Springer, Berlin (1992)CrossRefMATHGoogle Scholar
  2. 2.
    Bishop, R.L.: There is more than one way to frame a curve. Am. Math. Mon. 82, 246–251 (1975)MathSciNetCrossRefMATHGoogle Scholar
  3. 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
  4. 4.
    Bölcskei, A., Szilágyi, B.: Frenet formulas and geodesics in Sol geometry. Beitr. Algebra Geom. 48, 411–421 (2007)MathSciNetMATHGoogle Scholar
  5. 5.
    Cartan, E.: Leçons sur la géométrie des espaces de Riemann, 2ème edn. Gauthier-Villars, Paris (1946)MATHGoogle Scholar
  6. 6.
    Castrillón López, M., Fernández Mateos, V., Muñoz Masqué, J.: The equivalence problem of curves in a Riemannian manifold. Ann. Mat. 194, 343–367 (2015)MathSciNetCrossRefMATHGoogle Scholar
  7. 7.
    Castrillón López, M., Muñoz Masqué, J.: Invariants of Riemannian curves in dimensions 2 and 3. Differ. Geom. Appl. 35, 125–135 (2014)MathSciNetCrossRefMATHGoogle Scholar
  8. 8.
    Chen, B.Y.: When does the position vector of a space curve always lie in its rectifying plane? Am. Math. Mon. 110, 147–152 (2003)MathSciNetCrossRefMATHGoogle Scholar
  9. 9.
    Chen, B.Y.: Rectifying curves and geodesics on a cone in the Euclidean 3-space. Tamkang J. Math. 48, 1 (2017)MathSciNetCrossRefGoogle Scholar
  10. 10.
    Chen, B.Y.: Topics in differential geometry associated with position vector fields on Euclidean submanifolds. Arab J. Math. Sci. 23, 1–17 (2017)MathSciNetMATHGoogle Scholar
  11. 11.
    da Silva, L.C.B.: Characterization of spherical and plane curves using rotation minimizing frames (2017). arXiv:1706.01577v3
  12. 12.
    da Silva, L.C.B.: Moving frames and the characterization of curves that lie on a surface. J. Geom. 108, 1091 (2017)MathSciNetCrossRefMATHGoogle Scholar
  13. 13.
    do Carmo, M.P.: Riemannian Geometry. Birkhäuser, Boston (1992)CrossRefMATHGoogle Scholar
  14. 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
  15. 15.
    Etayo, F.: Geometric properties of rotation minimizing vector fields along curves in Riemannian manifolds. Turk. J. Math. 42, 121 (2018)CrossRefGoogle Scholar
  16. 16.
    Farouki, R.T.: Pythagorean-Hodograph Curves: Algebra and Geometry Inseparable. Springer, Berlin (2008)CrossRefMATHGoogle Scholar
  17. 17.
    Gökçelik, F., Bozkurt, Z., Gök, I., Ekmekci, F.N., Yaylı, Y.: Parallel transport frame in 4-dimensional Euclidean space E\(^4\). Caspian J. Math. Sci. 3, 91–103 (2014)MathSciNetGoogle Scholar
  18. 18.
    Guggenheimer, H.W.: Computing frames along a trajectory. Comput. Aided Geom. Des. 6, 77–78 (1989)MathSciNetCrossRefMATHGoogle Scholar
  19. 19.
    Gutkin, E.: Curvatures, volumes and norms of derivatives for curves in Riemannian manifolds. J. Geom. Phys. 61, 2147–2161 (2011)MathSciNetCrossRefMATHGoogle Scholar
  20. 20.
    Kreyszig, E.: Differential Geometry. Dover, New York (1991)MATHGoogle Scholar
  21. 21.
    Kreyszig, E., Pendl, A.: Spherical curves and their analogues in affine differential geometry. Proc. Am. Math. Soc. 48, 423–428 (1975)MathSciNetCrossRefMATHGoogle Scholar
  22. 22.
    Kühnel, W.: Differentialgeometrie: Kurven–Flächen–Mannigfaltigkeiten 5. Auflage. Vieweg+Teubner (2010)Google Scholar
  23. 23.
    Lucas, P., Ortega-Yagües, J.A.: Rectifying curves in the three-dimensional sphere. J. Math. Anal. Appl. 421, 1855–1868 (2015)MathSciNetCrossRefMATHGoogle Scholar
  24. 24.
    Lucas, P., Ortega-Yagües, J.A.: Rectifying curves in the three-dimensional hyperbolic space. Mediterr. J. Math. 13, 2199–2214 (2016)MathSciNetCrossRefMATHGoogle Scholar
  25. 25.
    Murphy, T., Wilhelm, F.: Random manifolds have no totally geodesic submanifolds. https://arxiv.org/abs/1703.09240 (To appear in Michigan Math. J.)
  26. 26.
    Nikolayevsky, Y.: Totally geodesic hypersurfaces of homogeneous spaces. Israel J. Math. 207, 361–375 (2015)MathSciNetCrossRefMATHGoogle Scholar
  27. 27.
    Pottmann, H., Wagner, M.: Contributions to motion based surface design. Int. J. Shape Model. 4, 183–196 (1998)CrossRefGoogle Scholar
  28. 28.
    Reynolds, W.F.: Hyperbolic geometry on a hyperboloid. Am. Math. Mon. 100, 442–455 (1993)MathSciNetCrossRefMATHGoogle Scholar
  29. 29.
    Siltanen, P., Woodward, C.: Normal orientation methods of 3D offset curves, sweep surfaces and skinning. Comput. Graph. Forum 11, 449–457 (1992)CrossRefGoogle Scholar
  30. 30.
    Spivak, M.: A comprehensive introduction to differential geometry, vol. 4, 2nd edn. Publish or Perish, Houston (1979)MATHGoogle Scholar
  31. 31.
    Szilágyi, B., Virosztek, D.: Curvature and torsion of geodesics in three homogeneous Riemannian 3-geometries. Stud. Univ. Žilina Math. Ser. 16, 1–7 (2003)MATHGoogle Scholar
  32. 32.
    Tsukada, K.: Totally geodesic submanifolds of Riemannian manifolds and curvature-invariant subspaces. Kodai Math. J. 19, 395–437 (1996)MathSciNetCrossRefMATHGoogle Scholar
  33. 33.
    Wang, W., Jüttler, B., Zheng, D., Liu, Y.: Computation of rotation minimizing frames. ACM Trans. Graph. 27, Article 2 (2008)Google Scholar
  34. 34.
    Wong, Y.: A global formulation of the condition for a curve to lie on a sphere. Monatsh. Math. 67, 363–365 (1963)MathSciNetCrossRefMATHGoogle Scholar

Copyright information

© Springer International Publishing AG, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Departamento de MatemáticaUniversidade Federal de PernambucoRecifeBrazil
  2. 2.Departamento de MatemáticaUniversidade Federal Rural de PernambucoRecifeBrazil

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