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Geometriae Dedicata

, Volume 57, Issue 2, pp 195–206 | Cite as

Counterexamples to isosystolic inequalities

  • Mikhail Katz
Article

Abstract

We explore M. Gromov's counterexamples to systolic inequalities. Does the manifoldS2 ×S2 admit metrics of arbitrarily small volume such that every noncontractible surface inside it has at least unit area? This question is still open, but the answer is affirmative for its analogue in the case ofS n ×S n ,n ≥ 3. Our point of departure is M. Gromov's metric onS1 ×S3, and more general examples, due to C. Pittet, of metrics onS1 ×S n with ‘voluminous’ homology. We take the metric product of these metrics with a sphereSn−1 of a suitable volume, and perform surgery to obtain the desired metrics onS n ×S n .

Mathematics Subject Classifications (1991)

53C23 

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Copyright information

© Kluwer Academic Publishers 1995

Authors and Affiliations

  • Mikhail Katz
    • 1
  1. 1.Département de MathématiquesUniversité de Nancy 1VandoeuvreFrance

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