Abstract
We explore M. Gromov's counterexamples to systolic inequalities. Does the manifoldS 2 ×S 2 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 onS 1 ×S 3, and more general examples, due to C. Pittet, of metrics onS 1 ×S n with ‘voluminous’ homology. We take the metric product of these metrics with a sphereS n−1 of a suitable volume, and perform surgery to obtain the desired metrics onS n ×S n.
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Katz, M. Counterexamples to isosystolic inequalities. Geom Dedicata 57, 195–206 (1995). https://doi.org/10.1007/BF01264937
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DOI: https://doi.org/10.1007/BF01264937