## Abstract

We construct metrics X_{n} of arbitrarily small volume on the product S^{1}×S^{2} of the circle and the 2-sphere, verifying the inequality length(γ)·area(Σ)≥1, for every noncontractible curve γ and every homologically nontrivial surface Σ in S^{1}×S^{2} (a product metric cannot have such a property). In other words, the intersystolic inequality for S^{1}×S^{2} is false (in contrast with the stable intersystolic inequality). We construct the metrics X_{n}, which violate the inequality, by surgery on a left-invariant metric on the Heisenberg group. The X_{n} are Riemannian submersions of cohomogeneity +1, whose fibers circle n times a closed loop in the moduli space of flat tori. The X_{n} are inspired by M. Gromov's homogeneous metrics on S^{1}×S^{3} violating the intersystolic inequality. The X_{n} have an alternative description as "twisted" warped product metrics, where the S^{2}-factor is deformed following a certain flow, giving rise to a non-conformal deformation. We obtain a lower bound for the 2-systole of X_{n} either by calibration by a truncated area form, or by the use of an area-decreasing projection of the universal cover ∼X_{n}. The lower bound for the 1-systole relies on a suitable map to a 3-dimensional compact cell complex containing the standard nilmanifold of the Heisenberg group.

Original language | English |
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Pages (from-to) | 621-632 |

Number of pages | 12 |

Journal | Geometric and Functional Analysis |

Volume | 4 |

Issue number | 6 |

DOIs | |

State | Published - Nov 1994 |

Externally published | Yes |