Advertisement

Geometriae Dedicata

, Volume 155, Issue 1, pp 105–140 | Cite as

Extrinsic diameter of immersed flat tori in S 3

  • Yoshihisa Kitagawa
  • Masaaki UmeharaEmail author
Original Paper

Abstract

Enomoto, Weiner and the first author showed the rigidity of the Clifford torus amongst the class of embedded flat tori in S 3. In the proof of that result, an estimate of extrinsic diameter of flat tori plays a crucial role. It is reasonable to expect that the same rigidity holds in the class of immersed flat tori in S 3. In this paper, we give a new method for characterizing immersed flat tori in S 3 with extrinsic diameter π, which is a somewhat similar technique to the proof of the 6-vertex theorem for certain closed plane curves given by the second author. As an application, we show that the Clifford torus is rigid in the class of immersed flat tori whose mean curvature functions do not change sign. Recently, the global behaviour of flat surfaces in H 3 and R 3 regarded as wave fronts has been studied. We also give here a formulation of flat tori in S 3 as wave fronts. As an application, we shall exhibit a flat torus as a wave front whose extrinsic diameter is less than π.

Keywords

Rigidity Flat torus Clifford torus Wave front Front 3-sphere Mean curvature Gaussian curvature Extrinsic diameter 

Mathematics Subject Classification (2000)

53C45 53C40 53C42 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Arnold, V.I.: Topological Invariants of Plane Curves and Caustics. University Lecture Series 5, American Mathmatical Society, Providence, R.I. (1994)Google Scholar
  2. 2.
    Bianchi L.: Sulle superficie a curvatura nulla in geometria ellittica. Ann. Mat. Pura Appl. 24, 93–129 (1896)CrossRefzbMATHGoogle Scholar
  3. 3.
    Enomoto K., Kitagawa Y., Weiner J.L.: A rigidity theorem for the Clifford tori in S 3. Proc. Am. Math. Soc. 124, 265–268 (1996)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Gálvez J.A., Mira P.: Isometric immersions of R 2 into R 4 and perturbation of Hopf tori. Math. Z. 266, 207–227 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Jackson S.B.: Vertices of plane curves. Bull. Am. Math. Soc. 50, 564–578 (1944)CrossRefzbMATHGoogle Scholar
  6. 6.
    Kitagawa Y.: Periodicity of the asymptotic curves on flat tori in S 3. J. Math. Soc. Jpn. 40, 457–476 (1988)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Kitagawa Y.: Embedded flat tori in the unit 3-sphere. J. Math. Soc. Jpn. 47(2), 275–296 (1995)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Kitagawa Y.: Flat tori in the 3-dimensional sphere. Sugaku Expos. 21, 133–145 (2008)MathSciNetzbMATHGoogle Scholar
  9. 9.
    Kneser, H.: Neuer Beweis des Vierscheitelsatzes. Christiaan Huygens 2, 315–318 (1922/1923)Google Scholar
  10. 10.
    Kobayashi O., Umehara M.: Geometry of scrolls. Osaka J. Math. 33, 441–473 (1996)MathSciNetzbMATHGoogle Scholar
  11. 11.
    Kurono, Y., Umehara, M.: Flat Möbius strips of given isotopy type in R 3 whose centerline are geodesic or lines of curvature. Geom. Dedicata 109–130 (2008)Google Scholar
  12. 12.
    Kokubu M., Rossman W., Saji K., Umehara M., Yamada K.: Singularities of flat fronts in hyperbolic 3-space. Pac. J. Math. 221, 303–351 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Kokubu M., Rossman W., Umehara M., Yamada K.: Flat fronts in hyperbolic 3-space and their caustics. J. Math. Soc. Jpn. 59, 265–299 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Leon-Guzman, M.A., Mira, P., Pastor, J.A.: The space of Lorentzian flat tori in anti-de Sitter 3-space (to appear in Trans. Am. Math. Soc. arXiv:0905.3991)Google Scholar
  15. 15.
    Murata S., Umehara M.: Flat surfaces with singularities in Euclidean 3-space. J. Diff. Geom. 82, 279–316 (2009)MathSciNetzbMATHGoogle Scholar
  16. 16.
    Okada, T.: Flat fronts in S 3 (in Japanese). Master’s Thesis, Osaka University (2005)Google Scholar
  17. 17.
    Pikall U.: Hopf tori in S 3. Invent Math. 81, 21–37 (1985)Google Scholar
  18. 18.
    Roitman P.: Flat surfaces in hyperbolic 3-space as normal surfaces to a congruence of geodesics. Tohoku Math. J. 59, 21–37 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Saji K., Umehara M., Yamada K.: Geometry of fronts. Ann. Math. 169, 491–529 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Saji K., Umehara M., Yamada K.: Coherent tangent bundles and Gauss-Bonnet formulas for wave fronts. J. Geom. Anal. 12, 1–27 (2010)Google Scholar
  21. 21.
    Saji, K., Umehara, M., Yamada, K.: A 2-singularities of hypersurfaces with non-negative sectional curvature in Euclidean space, preprint, arXiv:1011.1544Google Scholar
  22. 22.
    Thorbergsson G., Umehara M.: Inflection points and double tangents on anti-convex curves in the real projective plane. Tohoku Math. J. 60, 149–181 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Umehara M.: 6-vertex theorem for closed planar curve which bounds an immersed surface with non-zero genus. Nagoya Math. J. 134, 75–89 (1994)MathSciNetzbMATHGoogle Scholar
  24. 24.
    Umehara, M.: A Unified Approach to the Four Vertex Theorems I. Differential and Symplectic Topology of Knots and Curves, pp. 185–228. American Mathematical Society Translation Series 2 No. 190, American Mathematical Society, Providence, R.I (1999)Google Scholar
  25. 25.
    Umehara, M., Yamada, K.: Applications of a completeness lemma in minimal surface theory to various classes of surfaces. (to appear in Bull. Lond. Math. Soc.) arXiv:0909.1128Google Scholar
  26. 26.
    Weiner J.L.: Flat tori in S 3 and their Gauss maps. Proc. Lond. Math. Soc. 62(3), 54–76 (1991)MathSciNetCrossRefzbMATHGoogle Scholar
  27. 27.
    Wunderlich W.: Über ein abwickelbares Möbiusband. Monatsh. Math. 66, 276–289 (1962)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer Science+Business Media B.V. 2011

Authors and Affiliations

  1. 1.Department of MathematicsUtsunomiya UniversityUtsunomiyaJapan
  2. 2.Department of Mathematics, Graduate School of ScienceOsaka UniversityToyonaka, OsakaJapan

Personalised recommendations