TY - JOUR

T1 - An optimal Loewner-type systolic inequality and harmonic one-forms of constant norm

AU - Bangert, Victor

AU - Katz, Mikhail

PY - 2004/7

Y1 - 2004/7

N2 - 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, AX, of X. For an extremal metric, the map AX turns out to be a Riemannian submersion with minimal fibers, onto a flat torus. We characterize the base of Ax in terms of an extremal problem for Euclidean lattices, studied by A.-M. Bergé and J. Martinet. Given a closed manifold X that admits a submersion F to its Jacobi torus Tb1(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.

AB - 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, AX, of X. For an extremal metric, the map AX turns out to be a Riemannian submersion with minimal fibers, onto a flat torus. We characterize the base of Ax in terms of an extremal problem for Euclidean lattices, studied by A.-M. Bergé and J. Martinet. Given a closed manifold X that admits a submersion F to its Jacobi torus Tb1(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.

UR - http://www.scopus.com/inward/record.url?scp=4544220180&partnerID=8YFLogxK

U2 - 10.4310/CAG.2004.v12.n3.a8

DO - 10.4310/CAG.2004.v12.n3.a8

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AN - SCOPUS:4544220180

SN - 1019-8385

VL - 12

SP - 703

EP - 732

JO - Communications in Analysis and Geometry

JF - Communications in Analysis and Geometry

IS - 3

ER -