Geometriae Dedicata

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

Hyperellipticity and systoles of Klein surfaces

  • Mikhail G. KatzEmail author
  • Stéphane Sabourau
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 


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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

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