Geometriae Dedicata

, Volume 121, Issue 1, pp 61–71 | Cite as

(On the systole of the sphere in the proximity of the standard metric)

  • Florent BalacheffEmail author
Original Paper


We study the systolic area (defined as the ratio of the area over the square of the systole) of the 2-sphere endowed with a smooth Riemannian metric as a function of this metric. This function, bounded from below by a positive constant over the space of metrics, admits the standard metric g 0 as a critical point, although it does not achieve the conjectured global minimum: we show that for each tangent direction to the space of metrics at g 0, there exists a variation by metrics corresponding to this direction along which the systolic area can only increase.


Critical point 2-sphere Standard metric Systole Zoll metric 

Sur la systole de la sphère au voisinage de la métrique standard

Mathematics Subject Classifications (2000)

53C22 58E10 37C27 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Besse A. (1978) Manifolds All of whose Geodesics are Closed. Springer, Berlin-Heidelberg, New YorkzbMATHGoogle Scholar
  2. 2.
    Calabi E., Cao J. (1992) Simple closed geodesics on convex surfaces. J. Differential Geom. 36, 517–549zbMATHMathSciNetGoogle Scholar
  3. 3.
    Croke, C.: Lower bounds on the energy of maps. Duke Math. J. 55(4), 901–908 (1987)Google Scholar
  4. 4.
    Croke C. (1988) Area and the length of the shortest closed geodesic. J. Differential Geom. 27: 1–21zbMATHMathSciNetGoogle Scholar
  5. 5.
    Funk P. (1913) Über Fläschen mit lauter geschlossenen geodätischen Linien. Math. Ann. 74, 278–300CrossRefMathSciNetGoogle Scholar
  6. 6.
    Guillemin V. (1976) The Radon transform on Zoll surfaces. Adv. Math. 22, 85–119zbMATHCrossRefMathSciNetGoogle Scholar
  7. 7.
    Klingenberg W. (1978) Lectures on Closed Geodesics. Springer, Berlin-Heidelberg, New YorkzbMATHGoogle Scholar
  8. 8.
    Milnor, J.: Remarks on infinite-dimensional Lie groups. Relativity, groups and topology, II (Les Houches, 1983), 1007–1057 (1984)Google Scholar
  9. 9.
    Nabutovsky, A., Rotman, R.: The length of the shortest closed geodesic on a 2-dimensional sphere. Int. Math. Res. Not. 23, 1211–1222 (2002)Google Scholar
  10. 10.
    Pu P. (1952) Some inequalities in certain nonorientable Riemannian manifolds. Pacific J. Math. 2, 55–72zbMATHMathSciNetGoogle Scholar
  11. 11.
    Rotman R.(2006). The length of a shortest closed geodesic and the area of a 2-dimensional sphere. Proc. Amer. Math. Soc. 134, 3041–3047zbMATHCrossRefMathSciNetGoogle Scholar
  12. 12.
    Sabourau S. (2004) Filling radius and short closed geodesics of the 2-sphere. Bull. SMF 132, 105–136zbMATHMathSciNetGoogle Scholar
  13. 13.
    Weinstein A. (1974) On the volume of manifolds all of whose geodesics are closed. J. Diff. Geom. 9, 513–517zbMATHMathSciNetGoogle Scholar
  14. 14.
    Zoll O. (1903) Über Flächen mit Scharen geschlossener geodätischer Linien. Math. Ann. 57, 108–133CrossRefMathSciNetGoogle Scholar

Copyright information

© Springer Science + Business Media B.V. 2006

Authors and Affiliations

  1. 1.Section de MathématiquesUniversité de GenèveGenèveSwitzerland

Personalised recommendations