Geometriae Dedicata

, Volume 159, Issue 1, pp 277–293 | Cite as

Hyperellipticity and systoles of Klein surfaces

Original Paper


Given a hyperelliptic Klein surface, we construct companion Klein bottles, extending our technique of companion tori already exploited by the authors in the genus 2 case. Bavard’s short loops on such companion surfaces are studied in relation to the original surface so to improve a systolic inequality of Gromov’s. A basic idea is to use length bounds for loops on a companion Klein bottle, and then analyze how curves transplant to the original non-orientable surface. We exploit the real structure on the orientable double cover by applying the coarea inequality to the distance function from the real locus. Of particular interest is the case of Dyck’s surface. We also exploit an optimal systolic bound for the Möbius band, due to Blatter.


Antiholomorphic involution Coarea formula Dyck’s surface Hyperelliptic curve Möbius band Klein bottle Riemann surface Klein surface Loewner’s torus inequality Systole 

Mathematics Subject Classification (2000)

53C23 30F10 58J60 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Ambrosio, L., Katz, M.: Flat currents modulo p in metric spaces and filling radius inequalities. Comment. Math. Helv. 86(3), 557–591 (2011). See and arXiv:1004.1374
  2. 2.
    Babenko, I., Balacheff, F.: Distribution of the systolic volume of homology classes. See arXiv:1009.2835Google Scholar
  3. 3.
    Balacheff, F., Parlier, H., Sabourau, S.: Short loop decompositions of surfaces and the geometry of Jacobians. Geom. Funct. Anal., (to appear). See arXiv:1011.2962Google Scholar
  4. 4.
    Bangert V., Croke C., Ivanov S., Katz M.: Filling area conjecture and ovalless real hyperelliptic surfaces. Geom. Funct. Anal. 15(3), 577–597 (2005) See arXiv:math.DG/0405583MathSciNetMATHCrossRefGoogle Scholar
  5. 5.
    Belolipetsky, M.: Geodesics, volumes and Lehmer’s conjecture. See arXiv:1106.1834Google Scholar
  6. 6.
    Bavard C.: Inégalité isosystolique pour la bouteille de Klein. Math. Ann. 274(3), 439–441 (1986)MathSciNetMATHCrossRefGoogle Scholar
  7. 7.
    Berger M.: A Panoramic View of Riemannian Geometry. Springer, Berlin (2003)MATHCrossRefGoogle Scholar
  8. 8.
    Berger M.: What is... a Systole?. Notices AMS 55(3), 374–376 (2008)MATHGoogle Scholar
  9. 9.
    Blatter C.: Über Extremallängen auf geschlossenen Flächen. Comment. Math. Helv. 35, 153–168 (1961)MathSciNetMATHCrossRefGoogle Scholar
  10. 10.
    Blatter C.: Zur Riemannschen Geometrie im Grossen auf dem Möbiusband. Compos. Math. 15, 88–107 (1961)MathSciNetMATHGoogle Scholar
  11. 11.
    Brunnbauer M.: Homological invariance for asymptotic invariants and systolic inequalities. Geom. Funct. Anal. 18(4), 1087–1117 (2008) See arXiv:math.GT/0702789MathSciNetMATHCrossRefGoogle Scholar
  12. 12.
    Brunnbauer M.: Filling inequalities do not depend on topology. J. Reine Angew. Math. 624, 217–231 (2008) See arXiv:0706.2790MathSciNetMATHCrossRefGoogle Scholar
  13. 13.
    Brunnbauer M.: On manifolds satisfying stable systolic inequalities. Math. Ann. 342(4), 951–968 (2008) See arXiv:0708.2589MathSciNetMATHCrossRefGoogle Scholar
  14. 14.
    Dranishnikov A., Katz M., Rudyak Y.: Small values of the Lusternik-Schnirelmann category for manifolds. Geom. Topol. 12, 1711–1727 (2008) See arXiv:0805.1527MathSciNetMATHCrossRefGoogle Scholar
  15. 15.
    Dranishnikov A., Katz M., Rudyak Y.: Cohomological dimension, self-linking, and systolic geometry. Israel J. Math. 184(1), 437–453 (2011) See arXiv:0807.5040MathSciNetCrossRefGoogle Scholar
  16. 16.
    El Mir, C.: Conformal isosystolic inequality of Bieberbach 3-manifolds. See arXiv:1007.0877Google Scholar
  17. 17.
    Farkas H.M., Kra I.: Riemann Surfaces 2nd edn. Graduate Texts in Mathematics, vol. 71. Springer, New York (1992)Google Scholar
  18. 18.
    Fetaya, E.: Homological Error Correcting Codes and Systolic Geometry. See
  19. 19.
    Freedman M., Hass J., Scott P.: Closed geodesics on surfaces. Bull. Lond. Math. Soc. 14(5), 385–391 (1982)MathSciNetMATHCrossRefGoogle Scholar
  20. 20.
    Gendulphe, M.: Paysage Systolique Des Surfaces Hyperboliques Compactes De Caracteristique −1. See arXiv:math/0508036Google Scholar
  21. 21.
    Gromov M.: Filling Riemannian manifolds. J. Differ. Geom. 18, 1–147 (1983)MathSciNetMATHGoogle Scholar
  22. 22.
    Gromov, M.: Systoles and intersystolic inequalities. Actes de la Table Ronde de Géométrie Différentielle (Luminy, 1992), 291–362, Sémin. Congr., vol. 1, Soc. Math. France, Paris, (1996).
  23. 23.
    Gromov M.: Metric Structures for Riemannian and Non-Riemannian Spaces. Progr. Math vol. 152. Birkhäuser, Boston (1999)Google Scholar
  24. 24.
    Gromov, M.: Metric structures for Riemannian and non-Riemannian spaces. Based on the 1981 French original. With appendices by M. Katz, P. Pansu and S. Semmes. Translated from the French by Sean Michael Bates. Reprint of the 2001 English edition. Modern Birkhäuser Classics. Birkhäuser Boston, Inc., Boston, MA (2007)Google Scholar
  25. 25.
    Guth L.: Systolic inequalities and minimal hypersurfaces. Geom. Funct. Anal. 19(6), 1688–1692 (2010) See arXiv:0903.5299MathSciNetMATHCrossRefGoogle Scholar
  26. 26.
    Guth L.: Volumes of balls in large Riemannian manifolds. Ann. Math. 173(1), 51–76 (2011) See arXiv:math.DG/0610212MathSciNetMATHCrossRefGoogle Scholar
  27. 27.
    Horowitz C., Katz K., Katz M.: Loewner’s torus inequality with isosystolic defect. J. Geom. Anal. 19(4), 796–808 (2009) See arXiv:0803.0690MathSciNetMATHCrossRefGoogle Scholar
  28. 28.
    Katz K., Katz M.: Bi-Lipschitz approximation by finite-dimensional imbeddings. Geom. Dedic. 150(1), 131–136 (2010) Available at the site arXiv:0902.3126CrossRefGoogle Scholar
  29. 29.
    Katz K., Katz M., Sabourau S., Shnider S., Weinberger Sh.: Relative systoles of relative-essential 2-complexes. Algebraic Geom. Topol. 11, 197–217 (2011) See arXiv:0911.4265MathSciNetMATHCrossRefGoogle Scholar
  30. 30.
    Katz, M.: Systolic geometry and topology. With an appendix by Jake P. Solomon. Mathematical Surveys and Monographs, vol. 137. American Mathematical Society, Providence, RI (2007)Google Scholar
  31. 31.
    Katz M.: Systolic inequalities and Massey products in simply-connected manifolds. Israel J. Math. 164, 381–395 (2008) arXiv:math.DG/0604012MathSciNetMATHCrossRefGoogle Scholar
  32. 32.
    Katz M., Sabourau S.: Entropy of systolically extremal surfaces and asymptotic bounds. Ergo. Th. Dyn. Syst. 25(4), 1209–1220 (2005) See arXiv:math.DG/0410312MathSciNetMATHCrossRefGoogle Scholar
  33. 33.
    Katz M., Sabourau S.: Hyperelliptic surfaces are Loewner. Proc. Am. Math. Soc. 134(4), 1189–1195 (2006) See arXiv:math.DG/0407009MathSciNetMATHCrossRefGoogle Scholar
  34. 34.
    Katz M., Schaps M., Vishne U.: Logarithmic growth of systole of arithmetic Riemann surfaces along congruence subgroups. J. Differ. Geom. 76(3), 399–422 (2007)MathSciNetMATHGoogle Scholar
  35. 35.
    Katz, M., Schaps, M., Vishne, U.: Hurwitz quaternion order and arithmetic Riemann surfaces. Geom. Dedic. (2011). Online first and arXiv:math.RA/0701137
  36. 36.
    Katz, M., Shnider, S.: Cayley 4-form comass and triality isomorphisms. Israel J. Math. (2010), to appear. See arXiv:0801.0283Google Scholar
  37. 37.
    Lelièvre, S., Silhol, R.: Multi-geodesic tessellations, fractional Dehn twists and uniformization of algebraic curves. See
  38. 38.
    Makover E., McGowan J.: The length of closed geodesics on random Riemann surfaces. Geom. Dedic. 151, 207–220 (2011) See arXiv:math.DG/0504175MathSciNetMATHCrossRefGoogle Scholar
  39. 39.
    Miranda, R.: Algebraic curves and Riemann surfaces. Graduate Studies in Mathematics, vol. 5. American Mathematical Society, Providence, RI (1995)Google Scholar
  40. 40.
    Parlier H.: Fixed-point free involutions on Riemann surfaces. Israel J. Math. 166, 297–311 (2008) See arXiv:math.DG/0504109MathSciNetMATHCrossRefGoogle Scholar
  41. 41.
    Parlier, H.: The homology systole of hyperbolic Riemann surfaces. Geom. Dedic. (2011). see and arXiv:1010.0358
  42. 42.
    Pu P.M.: Some inequalities in certain nonorientable Riemannian manifolds. Pac. J. Math. 2, 55–71 (1952)MathSciNetMATHGoogle Scholar
  43. 43.
    Ryu H.: Stable systolic category of the product of spheres. Algebraic Geom. Topol. 11, 983–999 (2011) See arXiv:1007.2913MathSciNetMATHCrossRefGoogle Scholar
  44. 44.
    Sabourau S.: Isosystolic genus three surfaces critical for slow metric variations. Geom. Topol. 15, 1477–1508 (2011)MathSciNetMATHCrossRefGoogle Scholar
  45. 45.
    Sakai T.: A proof of the isosystolic inequality for the Klein bottle. Proc. Am. Math. Soc. 104(2), 589–590 (1988)MATHCrossRefGoogle Scholar
  46. 46.
    Silhol R.: On some one parameter families of genus 2 algebraic curves and half twists. Comment. Math. Helv. 82(2), 413–449 (2007)MathSciNetMATHCrossRefGoogle Scholar
  47. 47.
    Silhol, R.: Actions of fractional Dehn twists on moduli spaces. Geometry of Riemann surfaces, 376–395. Lond. Math. Soc. Lecture Note Ser. vol. 368. Cambridge University Press, Cambridge (2010)Google Scholar
  48. 48.
    Usadi K.: A counterexample to the equivariant simple loop conjecture. Proc. Am. Math. Soc. 118(1), 321–329 (1993)MathSciNetMATHCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media B.V. 2011

Authors and Affiliations

  1. 1.Department of MathematicsBar Ilan UniversityRamat GanIsrael
  2. 2.Laboratoire d’Analyse et Mathématiques Appliquées, UMR 8050Université Paris-EstCréteilFrance

Personalised recommendations