Geometriae Dedicata

, Volume 150, Issue 1, pp 131–136 | Cite as

Bi-Lipschitz approximation by finite-dimensional imbeddings

  • Karin Usadi Katz
  • Mikhail G. KatzEmail author
Original Paper


Gromov’s celebrated systolic inequality from ’83 is a universal volume lower bound in terms of the least length of a noncontractible loop in M. His proof passes via a strongly isometric imbedding called the Kuratowski imbedding, into the Banach space of bounded functions on M. We show that the imbedding admits an approximation by a \({(1+\epsilon)}\)-bi-Lipschitz (onto its image), finite-dimensional imbedding for every \({\epsilon > 0}\), using the first variation formula and the mean value theorem.


Essential manifold Finite-dimensional approximation First variation formula Geodesic Gromov’s inequality Infinitesimal Injectivity radius Kuratowski imbedding Systole Systolic inequality 

Mathematics Subject Classification (2000)

Primary 53C23 Secondary 26E35 


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. Commentarii Math. Helvetici(to appear). Avaiable at arXiv:1004.1374Google Scholar
  2. 2.
    Bangert V., Katz M., Shnider S., Weinberger S.: E 7, Wirtinger inequalities, Cayley 4-form, and homotopy. Duke Math. J. 146(1), 35–70 (2009) See arXiv:math.DG/0608006MathSciNetzbMATHCrossRefGoogle Scholar
  3. 3.
    Berger M.: What is... a Systole?. Notices AMS 55(3), 374–376 (2008)zbMATHGoogle Scholar
  4. 4.
    Brunnbauer M.: Homological invariance for asymptotic invariants and systolic inequalities. Geom. Funct. Anal. (GAFA) 18(4), 1087–1117 (2008) (See arXiv:math.GT/0702789)MathSciNetzbMATHCrossRefGoogle Scholar
  5. 5.
    Brunnbauer M.: Filling inequalities do not depend on topology. J. Reine Angew. Math. 624, 217–231 (2008) (See arXiv:0706.2790)MathSciNetzbMATHCrossRefGoogle Scholar
  6. 6.
    Brunnbauer M.: On manifolds satisfying stable systolic inequalities. Math. Ann. 342(4), 951–968 (2008) (See arXiv:0708.2589)MathSciNetzbMATHCrossRefGoogle Scholar
  7. 7.
    Dranishnikov A., Rudyak Y.: Stable systolic category of manifolds and the cup-length. J. Fixed Point Theory Appl. 6(1), 165–177 (2009)MathSciNetzbMATHCrossRefGoogle Scholar
  8. 8.
    Dranishnikov A., Katz M., Rudyak Y.: Small values of the Lusternik-Schnirelmann category for manifolds. Geom. Topol. 12, 1711–1727 (2008) (See arXiv:0805.1527)MathSciNetzbMATHCrossRefGoogle Scholar
  9. 9.
    Elmir C., Lafontaine J.: Sur la géométrie systolique des variétés de Bieberbach. Geom. Dedicata 136, 95–110 (2008) (See arXiv:0804.1419)MathSciNetCrossRefGoogle Scholar
  10. 10.
    Goldblatt R.: Lectures on the Hyperreals, An Introduction to Nonstandard Analysis, Graduate Texts in Mathematics, vol. 188. Springer, New York (1998)Google Scholar
  11. 11.
    Gromov M.: Filling Riemannian manifolds. J. Diff. Geom. 18, 1–147 (1983)MathSciNetzbMATHGoogle Scholar
  12. 12.
    Guth L.: Notes on Gromov’s systolic estimate. Geom. Dedicata 123, 113–129 (2006)MathSciNetzbMATHCrossRefGoogle Scholar
  13. 13.
    Guth L.: Systolic inequalities and minimal hypersurfaces. Geom. Funct. Anal. 19(6), 1688–1692 (2010) (See arXiv:0903.5299)MathSciNetzbMATHCrossRefGoogle Scholar
  14. 14.
    Guth, L.: Metaphors in systolic geometry. See arXiv:1003.4247Google Scholar
  15. 15.
    Guth, L.: Volumes of balls in large Riemannian manifolds. Annals of Mathematics, to appear. See arXiv:math.DG/0610212Google Scholar
  16. 16.
    Horowitz C., Katz K., Katz M.: Loewner’s torus inequality with isosystolic defect. J. Geom. Anal. 19(4), 796–808 (2009) (See arXiv:0803.0690)MathSciNetzbMATHCrossRefGoogle Scholar
  17. 17.
    Katsuda A.: Gromov’s convergence theorem and its application. Nagoya Math. J. 100, 11–48 (1985)MathSciNetzbMATHGoogle Scholar
  18. 18.
    Katz, K., Katz, M.: Hyperellipticity and Klein bottle companionship in systolic geometry. See arXiv:0811.1717Google Scholar
  19. 19.
    Katz, K., Katz, M., Sabourau, S., Shnider, S., Weinberger, Sh.: Relative systoles of relative-essential 2-complexes. See arXiv:0911.4265Google Scholar
  20. 20.
    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
  21. 21.
    Katz M.: Systolic inequalities and Massey products in simply-connected manifolds. Israel J. Math. 164, 381–395 (2008) (Available at the site arXiv:math.DG/0604012)MathSciNetzbMATHCrossRefGoogle Scholar
  22. 22.
    Katz, M., Shnider, S.: Cayley 4-form comass and triality isomorphisms. Israel J. Math., to appear. See arXiv:0801.0283 (2010)Google Scholar
  23. 23.
    Keisler H.J.: Elementary Calculus: an Infinitesimal Approach. 2nd edn. Prindle, Weber & Schimidt, Boston (1986)zbMATHGoogle Scholar
  24. 24.
    Keisler, H.J.: The hyperreal line. Real numbers, generalizations of the reals, and theories of continua, Synthese Lib., vol. 242, Kluwer Acad. Publ., Dordrecht, pp. 207–237 (1994)Google Scholar
  25. 25.
    Laugwitz D.: Definite values of infinite sums: aspects of the foundations of infinitesimal analysis around 1820. Arch. Hist. Exact Sci. 39(3), 195–245 (1989)MathSciNetzbMATHCrossRefGoogle Scholar
  26. 26.
    Lightstone A.H.: Infinitesimals. Am. Math. Monthly 79, 242–251 (1972)MathSciNetzbMATHCrossRefGoogle Scholar
  27. 27.
    Rudyak Y., Sabourau S.: Systolic invariants of groups and 2-complexes via Grushko decomposition. Ann. Inst. Fourier 58(3), 777–800 (2008) (See arXiv:math.DG/0609379)MathSciNetzbMATHCrossRefGoogle Scholar
  28. 28.
    Sabourau S.: Asymptotic bounds for separating systoles on surfaces. Comment. Math. Helv. 83(1), 35–54 (2008)MathSciNetzbMATHCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media B.V. 2010

Authors and Affiliations

  1. 1.Department of MathematicsBar Ilan UniversityRamat GanIsrael

Personalised recommendations