Counterexamples to isosystolic inequalities
In the present paper the author explores M. Gromov's counterexamples to systolic inequalities. Does the manifold \(S^2\times S^2\) admit metrics of arbitrarily small volume such that every noncontractible surface inside has at least unit area? This question is still open, but the author proves that the answer is affirmative for its analogue in the case of \(S^n\times S^n\), \(n\geq 3\). More precisely, combining some results of M. Gromov on \(S^1\times S^3\) and some results of C. Pittet on \(S^1\times S^n\), the author proves the following result: The manifold \(S^n\times S^n\) admits for all \(n\geq 3\) metrics with arbitrarily small ratio \(\text{vol/sys}^2_n\).
scientific article; zbMATH DE number 806282
scientific article