An optimal Loewner-type systolic inequality and harmonic one-forms of constant norm

Details An optimal Loewner-type systolic inequality and harmonic one-forms of constant norm An optimal Loewner-type systolic inequality and harmonic one-forms of constant norm

2003-04-30

arxiv.org/abs/math/0304494v1

Mathematics

Differential Geometry

27 pages

Scientific paper

We present a new optimal systolic inequality for a closed Riemannian manifold X, which generalizes a number of earlier inequalities, including that of C. Loewner. We characterize the boundary case of equality in terms of the geometry of the Abel-Jacobi map, A_X, of X. For an extremal metric, the map A_X turns out to be a Riemannian submersion with minimal fibers, onto a flat torus. We characterize the base of J_X in terms of an extremal problem for Euclidean lattices, studied by A.-M. Berg\'e and J. Martinet. Given a closed manifold X that admits a submersion F to its Jacobi torus T^{b_1(X)}, we construct all metrics on X that realize equality in our inequality. While one can choose arbitrary metrics of fixed volume on the fibers of F, the horizontal space is chosen using a multi-parameter version of J. Moser's method of constructing volume-preserving flows.

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